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HARVARD UNIVERSITY
PHYSICS RESEARCH
LIBRARY
Gift of J. H. Van Vleck
(^j^.^KwTl
A COURSE OF
MODERN ANALYSIS
lonlion: C. J. OLAY and SONS,
CAMBRIDGE UNIVERSITY PRESS WAREHOUSE,
AYE MARU LANE.
•UuwoiD: 60, WELLINGTON STREET.
lrtp}tg: F. A. BBOCKHAUS.
^ciD «odt: THE MAOMILLAN GOMPANT.
SombSB and ColnitU: MAOMILLAN AND 00.. Ltd.
[All RighU reserved.]
lS7T>A(.Z,
A COURSE OF
MODERN ANALYSIS
AN INTRODUCTION TO THE GENERAL THEORY OF
INFINITE SERIES AND OF ANALYTIC FUNCTIONS;
WITH AN ACCOUNT OF THE PRINCIPAL
TRANSCENDENTAL FUNCTIONS
BY
K T. WHITTAKER, M.A,
FELLOW AND LECTURER OF TRINITY COLLEGE, CAMBRIDGE
CAMBRIDGE:
AT THE UNIVERSITY PRESS.
1902
loo
Cambrttgr :
PRINTBD BT J. AHD 0. F. CLAT,
AT THE UNIVBR8ITT PRR88.
Physics Research Librar>
Jefferson Laboratoiy
Harvard University
MAR 19 1982
-?"'- '
V
y
PREFACE.
The first half -of this book contains an account of those methods and
processes of higher mathematical analysis, which seem to be of greatest
importance at the present time ; as will be seen by a glance at the table
of contents, it is chiefly concerned with the properties of infinite series
and complex integrals, and their applications to the analytical expression
of functions. A discussion of infinite determinants and of asymptotic
expansions has been included, as it seemed to be called for by the value of
these theories in connexion with linear differential equations and astronomy.
In the second half of the book, the methods of the earlier part are
applied in order to furnish the theory of the principal functions of analysis —
the Gamma, Legendre, Bessel, Hypergeometric, and Elliptic Functions. An
account has also been given of those solutions of the partial differential
equations of mathematical physics, which can be constructed by the help
of these functions.
My grateful thanks are due to two members of Trinity College,
Rev. E. M. Radford, M.A. (now of St John's School, Leatherhead), and
Mr J. E. Wright, B.A., who with great kindness and care have read the
proof-sheets; and to Professor Forsyth, for many helpful consultations
during the progress of the work. My great indebtedness to Dr Hobson*s
memoirs on Legendre functions must be specially mentioned here; and
I must thank the staff of. the University Press for their excellent co-
operation in the production of the volume.
E. T. WHITTAKER.
Cambridge,
1902 August 5
CONTENTS.
PART I. THE PROCESSES OF ANALYSIS.
CHAPTER I.
COMPLEX NUMBERS.
8KCTI0N PAGE
3
4
1. Heal numbers
2. Complex uumbers
3. The modulus of a complex quantity 5
4. The geometrical interpretation of complex numbers 6
Miscellaneous Examples 7
CHAPTER II.
THE THEORY OF ABSOLUTE CONVERGENCE.
5. The limit of a sequence of quantities 8
6. The necessary and sufficient conditions fi)r the existence of a limit . . 8
7. Convergence of an infinite series 10
8. Absolute convergence and semi-convergence 12
9. The geometric series, and the series 2n~* 13
10. The comparison-theorem 14
11. Discussion of a special series of importance 16
12. A convergency-test which depends on the ratio of the successive terms
of a series 17
13. A general theorem on those series for which Limit ( -* ) is 1 . . . 18
n-ao \u^ J
14. Convergence of the hypergeometric series 20
15. Effect of changing the order of the terms in a series 21
16. The fundamental property of absolutely convergent series .... 22
17. Riemann's theorem on semi-convergent series 22
18. Cauch/s theorem on the multiplication of absolutely convergent series . . 24
19. Mertens' theorem on the multiplication of a semi-convergent series by an
absolutely convergent series 25
20. Abel's result on the multiplication of series 26
21. Power-series 28
X CONTENTS.
SECTION PAGE
22. Convergence of series derived from a power-series 30
23. Infinite products 31
24. Some examples of infinite products 32
25. Cauchy's theorem on products which are not absolutely convergent . . 34
26. Infinite determinants 35
27. Convergence of an infinite determinant 36
28. Persistence of convergence when the elements are changed .... 37
Miscellaneous Examples 37
CHAPTER III.
THE FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS;
TAYLOR'S, LAURENT'S, AND LIOUVILLE'S THEOREMS.
29. The dependence of one complex number on another 40
30. Continuity 41
31. Definite integrals 42
32. Limit to the value of a definite integral 44
33. Property of the elementary functions 44
34. Occasional failure of the property ; singularities 45
35. The analytic function 45
36. Cauchy's theorem on the integral of a Amotion round a contour . . 47
37. The value of a function at a point, expressed as an integral taken round
a contour enclosing the point 50
38. The higher derivates 51
39. Taylor's theorem 54
40. Forms of the remainder in Taylor's series 56
41. The process of continuation 57
42. The identity of a function 59
43. Laurent's theorem 60
44. The nature of the singularities of a one-valued function .... 63
45. The point at infinity 64
46. Many-valued functions 66
47. Liouville's theorem 69
48. Functions with no essential singularities 69
'Miscellaneous Examples 70
CHAPTER IV.
THE UNIFORM CONVERGENCE OF INFINITE SERIES.
■
49. Uniform convergence
50. Connexion of discontinuity with non-uniform convergence
51. Distinction between absolute and uniform convergence
52. Condition for uniform convergence
53. Integration of infinite series .
54. Differentiation of infinite series
55. Uniform convergence of power-series
Miscellaneous Examples
73
76
77
78
78
81
81
82
CONTENTS. XI
CHAPTER V.
THE THEORY OF RESIDUES; APPLICATION TO THE EVALUATION
OF REAL DEFINITE INTEGRALS.
SECTION PAGE
56. Residues 83
57. Evaliiation of real definite integrals 84
58. Evaluation of the definite integral of a rational function .... 91
59. Cauchy's int^ral 92
60. The number of roots of an equation contained within a contour . . 92
61. Connexion between the zeros of a function and the zeros of its derivate . 93
MiSCSLLAKBOUS EXAMPLES 94
CHAPTER VI.
THE EXPANSION OF FUNCTIONS IN INFINITE SERIES.
62. Darboux's formula 96
63. The Bemoullian numbers and the Bemoullian polynomials ... 97
64. The Maclaurin-Bemoidlian expansion ........ 99
65. Burmann's theorem 100
66. Teixeira's extended form of Burmann's theorem 102
67. Evaluation of the coefficients 103
68. Expansion of a function of a root of an equation, in terms of a parameter
occurring in the equation 105
69. Lagrange's theorem 106
70. Rouch^'s extension of Lagrange's theorem 108
71. Teixeira*s generalisation of Lagrange's theorem 109
72. Laplace's extension of Lagrange's theorem 109
73. A further generalisation of Taylor's theorem 110
74. The expansion of a function as a series of rational functions . . . Ill
75. Expansion of a function as an infinite product 114
76. Elxpansion of a periodic function as a series of cotangents . . . . 116
77. Expansion in inverse factorials 117
Miscellaneous Examples 119
CHAPTER VII.
FOURIER SERIES.
78. Definition of Fourier series ; nature of the region within which a Fourier
series converges 127
79. Values of the coefficients in terms of the sum of a Fourier series, when the
series converges at all points in a belt of finite breadth in the «-plane . . 130
80. Fourier's theorem 131
81. The representation of a function by Fourier series for ranges other than
to 27r 137
82. The sine and cosine series 138
83. Alternative proof of Fourier's theorem 140
84. Nature of the convergence of a Fourier series 147
85. Determination of points of discontinuity 161
86. The uniqueness of the Fourier expansion 162
Miscellaneous Examples 167
XU CONTENTS.
CHAPTER VIII.
ASYMPTOTIC EXPANSIONS.
SECTION PAOK
87. Simple example of an asymptotic expansion 163
88. Definition of an asymptotic expansion 164
89. Another example of an asymptotic expansion 165
90. Multiplication of asymptotic expansions 167
91. Integration of asymptotic expansions 168
92. Uniqueness of an asymptotic expansion 168
Miscellaneous Examples 169
PART II. TRANSCENDENTAL FUNCTIONS.
CHAPTER IX.
THE GAMMA-FUNCTION.
93. Definition of the Qamma-f unction, Euler's form 173
94. The Weierstrassian form for the Gamma-function . . . . . 174
95. The difference-equation satisfied by the Gamma-function . . . 176
96. Evaluation of a general class of infinite products 177
97. Connexion between the Gamma-fimction and the circular functions . 179
98. The multiplication-theorem of Gauss and Legendre 179
99. Expansions for the logarithmic derivates of the Gamma-function . . 180
100. Heine*s expression of F («) as a contour-integral 181
101. Expression of T(z) as a definite integral, whose path of integration
is real 183
102. Extension of the definite-integral expression to the case in which the
argument of the Gamma-fimction is negative 184
103. Gauss' expression of the logarithmic derivate of the Gamma-fimction as
a definite integral 185
104. Binet's expression of log F (e) in terms of a definite integral . . 186
105. The Eulerian int^ral of the first kind 189
106. Expression of the Eulerian integral of the first kind in terms of Gkunma-
functions 190
107. Evaluation of trigonometric integrals in terms of the Gamma-function 191
108. Dirichlet's multiple integrals 191
109. The asymptotic expansion of the logarithm of the Gamma-fiinction (Stirling's
series) 193
110. Asymptotic expansion of the Gamma-fimction 194
Miscellaneous Examples 195
CONTENXa xm
CHAPTER X.
LEGENDRE FUNCTIONS.
BKCTION PAGE
111. DefinitioD of Legendre polynomiab 204
112. Schlafli's integral for P„ ^ 206
113. Rodrigues' formula for the Legendre polynomials 206
114. Legendre's differential equation 206
115. The int^^ral-properties of the L^endre polynomials 207
116. Legendre functions 208
117. The recurrence-formulae 210
118. Evaluation of the integral-expression for P^{z\ as a power-series . . 213
119. Laplace's iut^pral-expression iot P^{z) 215
120. The Mehler-Dirichlet definite integral for /*„(«). . % . . . 218
121. Expansion of P^ {z) as a series of powers of 1/z 220
122. The Legendre functions of the second kind 221
123. Expansion of Q^isi) as a power-series 222
124. The recurrence-formulae for the Legendre function of the second kind . . 2^4
125. Laplace's int^;ral for the Legendre function of the second kind . . . 225
126. Relation between Pm(^) ^^^ Q%{^\ when n is an integer . . . 226
127. Expansion of {t-x)~^ as a series of Legendre polynomials . . 228
128. Neumann's expansion of an arbitrary function as a series of Legendre
polynomials 230
129. The associated functions P^ (z) and $»"•(«) 231
130. The definite integrals of the associated Legendre functions . . . 232
131. Expansion of P%^{z) as a definite integral of Laplace's type . . 233
132. Alternative expression of P^ {z) as a definite integral of Laplace's type . 234
133. The function C/ («) 235
MiSCBLLANBOUS EXAMPLES 236
CHAPTER XI.
HYPERGEOMETRIC FUNCTIONS.
134. The hypergeometric series 240
136. Value of the series /'(a, 6, c, 1) . 241
136. The differential equation satisfied by the hypergeometric series 242
137. The differential equation of the general hypergeometric function 242
138. The L^^ndre functions as a particular case of the hypergeometric function . 245
139. Transformations of the general hypergeometric function .... 246
140. The twenty-four particular solutions of the hypergeometric differential
equation 249
141. Relations between the particular solutions of the hypergeometric differential
equation 251
142. Solution of the general hypergeometric differential equation by a definite
int^;ral 253
143. Determination of the integral which represents P(*) 257
144. Evaluation of a double-contour int^;ral 259
145. Relations between contiguous hypergeometric functions .... 260
MiSCBLLANBOUS EXAMPLES 263
;
xiv CONTENTS.
CHAPTER Xll.
BESSEL FUNCTIONS.
SECTION PAGE
146. The Bessel coefficients 266
147. Bessel's differential equation 268
148. Bessel's equation as a case of the hjpergeometric equation . . 269
149. The general solution of Bessel's equation by Bessel functions whose order is
not necessarily an integer 272
150. The recurrence-formulae for the Bessel functions 274
151. Relation between two Bessel functions whose orders differ by an integer . 275
152. The roots of Bessel functions 277
153. Expression of the Bessel coefficients as trigonometric integrals . . 277
154. Extension of the integral-formula to the caae in which n is not an integer . 279
155. A second expression of J^^ (z) as a definite integral whose path of integration
is real 282
156. Hankel's definite-integral solution of Bessel's differential equation . . 283
157. Expression of J^^ (2), for all values of n and «, by an integral of Hankel's type 284
158. Bessel functions as a limiting case of Legendre functions .... 287
159. Bessel functions whose order is half an odd integer 288
160. Expression of J^ (z) in a form which furnishes an approximate value to J^^ (z)
for large real positive values of z 289
161. The asymptotic expansion of the Bessel functions 292
162. The second solution of Bessel's equation when the order is an integer . . 294
163. Neumann's expansion ; determination of the coefficients .... 299
164 Proof of Neumann's expansion 300
165. Schl5milch's expansion of an arbitrary functiou in terms of Bessel functions
of order zero 302
166. Tabulation of the Bessel functions 304
Miscellaneous Examples 304
CHAPTER XIIL
APPLICATIONS TO THE EQUATIONS OF MATHEMATICAL PHYSICS.
167. Introduction : illustration of the general method 309
168. Laplace's equation ; the general solution ; certain particular solutions . 311
169. The series-solution of Laplace's equation 314
170. Determination of a solution of Laplace's equation which satisfies given
boundary-conditions 315
171. Particular solutions of Laplace's equation which depend on Bessel functions 317
172. Solution of the equation ^ -I- ^jj-hr=0 318
92 ^ 92 7 32 y
173. Solution of the equation g^ + ^j-h-^ -I- r=0 319
Miscellaneous Examples 321
CONTENTS. XV
CHAPTER XIV.
THE ELLIPTIC FUNCTION ^(z),
8BCTI0N PAGE
174. Introduction 322
175. Definition of jf> («) 323
176. Periodicity, and other properties, of ^(z) 324
177. The period-parallelograms 324
178. Expression of the function fp (z) by means of an integral .... 325
179. The homogeneity of the function i> (z) 329
180. The addition- theorem for the function ip{z) 329
181. Another form of the addition theorem 332
182. The roots Ci, eg, 63 333
183. Addition of a half-period to the aigument o{ jp{z) 334
181 Integration of (cw:* + 46a73+6c^ + 4ci:a7+e)-* 336
185. Another solution of the integration-problem 336
186. Uniformisation of curves of genus unity 338
Miscellaneous Examples 340
CHAPTER XV.
THE ELLIPTIC FUNCTIONS sn z, en z, dn z,
187. Construction of a doubly-periodic function with two simple poles in each
period-parallelogram 342
188. Expression of the function f{z) by means of an integral .... 343
189. The function sn^ 345
190. The functions cm and dn^ 346
191. Expression of en z and dn z by means of integrals 347
192. The addition-theorem for the function dnz 348
193. The addition-theorems for the functions sn z and en ^ . . . 350
191 The constant K 351
195. The periodicity of the elliptic functions with respect to ^ . . 351
196. The constant K' 352
197. The periodicity of the elliptic functions with respect to K+iK' . . 353
198. The periodicity of the elliptic functions with respect to tX' . . . 353
199. The behaviour of the functions snr, en-?, dnz, at the point z=iK' . . 354
200. (General description of the functions sn z, en 2, dn z 355
201. A geometrical illustration of the functions snz, cnz, dnz. . . 356
202. Connexion of the function snz with the function ^{z) . . . . 356
203. Expansion of snz as a trigonometric series 357
Miscellaneous Examples 359
XVI CONTENTS.
CHAPTER XVI.
ELLIPTIC FUNCTIONS; GENERAL THEOREMS.
SBCnON PAQK
204. Relation betweeu the residues of an elliptic function .... 362
205. The order of an elliptic Amotion 362
206. Expression of any elliptic function in terms of fp {z) and fP' (z) . . 363
207. Relation between any two elliptic fimctions which admit the same peri<xls . 364
208. Relation between the zeros and poles of an elliptic function . . 365
209. The function f (2) 366
210. The quasi-periodicity of the function ((z) 367
211. Expression of an elliptic function, when the principal part of its expansion
at each of its singularities is given 367
212. The function a{z) 368
213. The quasi-periodicity of the function ^{z) 369
214. The integration of an elliptic function 372
215. Expression of an elliptic function whose zeros and poles are known . 372
Miscellaneous Examples 374
Index 377
J
PART I.
THE PROCESSES OF ANALYSIS.
W. A.
CHAPTER I.
Complex Numbers.
1. Real Numbers.
The idea of a set of numbers is derived in the first instance ifrom the
consideration of the set of positive integral numbers, or positive integers ;
that is to say, the numbers 1, 2, 3, 4, .... Positive integers have many
properties, which will be found in treatises on the Theory of Integral
Numbers; but at a very early stage in the development of mathematics
it was found that they are inadequate to express all the quantities occurring
in calculations ; and so this primitive number system has come to be
enlarged. In elementary Arithmetic, and in the arithmetical applications
of Algebra, several new classes of numbers are defined, namely rational
fractions such as ^, negative numbers such as —3, and irrational numbers
such as the number 1*414213..., which represents the square root of 2.
The object of the introduction of these extended types of number is
that we may express the result of performing the operations of addition,
subtraction, multiplicatioo, division, involution, and evolution, on all integral
numbers. Thus, the result of dividing the integer 1 by the integer 2 is
inexpressible until we introduce the idea of fractional numbers: and the
result of subtracting the integer 2 from the integer 1 is inexpressible until
we introduce the idea of negative numbers.
The totality of the numbers introduced up to this point is called the
aggregate of real numbers.
The extension of the idea of number, which has just been described, was not effected
without some opposition from the more conservative mathematicians. In the latter half
of the 18th centiuy, Maseres (1731—1824) and Frend (1767—1841) published works
on Algebra, Trigonometry, etc., in which the use of negative quantities was disallowed,
although Descartes had used them imrestrictedly more than a hundred years before.
1—2
4 THE PROCESSES OF ANALYSIS. [CHAP. L
2. Complex Numbei^s*.
If we attempt to perform the operations already named — multiplication,
etc. — on any of the real numbers thus recognised, we find that there is one
case in which the result of the operation cannot be expressed without the
introduction of yet another type of numbers. The case referred to is that
in which the operation of evolution is applied to a negative number, e.g. to
find the square root of — 2. To express the results of this and similar opera-
tions, we make use of a new number, deooted by the letter t; this is defined
as a quantity which satisfies the fundamental laws of algebra (Le. can be
combined with other numbers according to the associative, distributive,
and commutative laws) and has for its square the negative number — 1.
It is easily seen that all the quantities \\h\c\\ can be formed by com-
bining i with real numbers are of the form a + 6t, where a and 6 are real
numbers. A quantity a + 6i of this nature is called (after Gauss) a complex
nv/mber. Real numbers may be regarded as a particular case of complex
numbers, corresponding to a zero value of the quantity 6.
The complex quantity thus introduced may in the first instance be
regarded as formed by the association of the pair of real numbers a and
6; as the quantities a, 6, i are subject to the ordinary laws of algebra,
we obtain for the addition and multiplication of two complex numbers
a + hi and c + dt the formulae
(a + hi) + (c + di) = (o + c) +(6 + d)t,
(a + 6t) (c + di) = (oc — bd) + {ad + he) i.
But a complex number will usually be considered apart from its composition^
as an irresoluble entity. Regarded in this light, it satisfies the fundamental
laws of algebra ; so that if a, 6, c are complex numbers, we have
a + 6 a 6 + a,
ah = 6a,
{a-{'h)-{'C^a-{'{h-\'C\
ah ,c^a ,hc,
a (6 + c) = oi + oc.
It is found that the operations of multiplication, etc., when applied to
complex numbers, do not lead to numbers of any fi"esh type ; the complex
number will therefore for our purposes be taken as the most general type
of number.
The introduction of the complex number has led to many important
developments in mathematics. Functions which, when real variables only
* For the general theory of complex numbers, see Hankel, Theorie der eomplexen Zahlen-
systeme (Leipzig, 1867), and Stolz, VarUsungen Uber allgemeine Arithmetik IL (Leipzig, 1886).
2, 3] COMPLEX NUMBERS. 5
are considered, appear as essentially distinct, are seen to be connected when
complex variables are introduced : thus the circular functions are found to
be expressible in terms of exponential functions of a complex argument, by
the equations
cos 07 = 2 («^ + e-^),
sin a? = 2i (^** - ^"**)-
Again, many of the most important theorems of modem analysis are
not true if the quantities concerned are restricted to be real; thus, the
theorem that every algebraic equation of degree n has n roots is true in
general only when complex values of the roots are admitted.
Hamilton's quaternions furnish an example of a still further extension of the idea
of number. A quaternion
is formed from four real numbers Wy x, y, z, and four number-units 1, i^ j, k, in the same
way as the ordinary complex number .r+ty is formed from two real numbers ^, y,
and two number-units 1, i Quaternions however do not obey the commutative law of
multiplication.
3. The modulus of a complex quantity.
Let x+iy he & complex quantity; x and y being real numbers. Then
the positive square root of a^ + y^ 18 called the modulus of (x + yi), and is
written
\x + yi\.
Let us consider the complex number which is the sum of two known
complex numbers, a? + iy and u + iv. We have
(x + iy) + (u + iv) « (a? + w) + i (y + v).
The modulus of the sum of the two numbers is therefore
or {(aj« + y«) + (u» + v») + 2 (xu -h yv)]K
But
= (^ + y*) + (t*» + v") + 2 {(xu -^ yvY •¥ (xv - yu)«}*,
and this latter expression is greater than (or at least equal to)
(«" + y*) + (w' + i^)+2(xu + yv).
We have therefore
I r + iy I + 1 u + iy I > |(a7 + iy) + (u + iv) |,
or the modulus of the sum of two complex numbers cannot be greater than the
sum of their moduli; and in general it follows that the modulus of the sum
6 THE PROCESSES OF ANALYSIS. [CHAP. I.
of any number of complex quantities cannot be greater than the sum of their
moduli.
Let us consider next the complex number which is the product of two
known complex numbers x -k-iy and u + %v; we have
{x + iy) {u + iv) = {xu — yv) + 1 (aw + yu),
and therefore
I {x-\-iy){u + iv) I = [{xu - yvf + (a?w+ ye^)'}*
^\x-^iy\ \u'\-iv\.
The modulus of the prodiLCt of two complex quantities (and hence of any
number of complex quantities) is therefore equal to the product of their modvli.
4. The geometrical interpretation of complex numbers.
For many purposes it is useful to represent complex numbers by a
geometrical diagram, which may be done in the following way.
Take rectangular axes Ox, Oy, in a plane. Then a point P whose
coordinates referred to these axes are x, y, will be regarded as representing
the complex number a?+ iy. In this way, to every point of the plane there
corresponds some complex number; and conversely, to every possible complex
number there corresponds one and only one point of the plane.
The complex number x + iy may be denoted by a single letter z. The
point P is then called the representative point or affix of the value z ; we
shall also speak of the number z as being the affia of the point P.
If we denote (a^+ y*)* by r and tan~* (^ i by d, then r and are clearly
the radius vector and vectorial angle of the point P, referred to the origin
and axis Ox,
The representation of complex quantities thus afforded is often called the
Argand diagram*.
If Pi and Pa are the representative points corresponding to values Zi
and z^ respectively of z, then the point which represents the value Zi-{- z^ is
clearly the terminus of a line drawn fix)m Pi, equal and parallel to that
which joins the origin to Pj.
To find the point which represents the complex number z^z^, where Zx and
z^ are two given complex' numbers, we notice that if
z^ = ri (cos ^1 + i sin d,),
iTj = r, (cos ^a + 1 sin d,),
♦ J. R. Argand published it in 1806 ; it bad bowever previously been used by Gauss, and
by Caspar Wessel, who discussed it in a miemoir published in 1797 to the Danish Academy.
4] COMPLEX NUMBERS. 7
then by multiplication
z^Zi =s r^ra {cos (d, + 0^) + i sin (0i + ^,)}.
The point which represents the value ZiZ^ has therefore a radius vector
measured by the product of the radii vectores of Pi and Pa, and a vectorial
angle equal to the sum of the vectorial angles of Pi and P,.
Miscellaneous Examples.
1. Shew that the representative points of the complex numbers 1+4^ 2+7i, 3 + lOt,
areooUinear.
2. Shew that a parabola can be drawn to pass through the representative points of
the complex numbers
2+t, 4+4i, 6 + 9t, 8+16t, 10+26i.
3. Determine by aid of the Argand diagram the nth roots of unity ; and shew that the
number of primitive roots (roots the powers of each of which give all the roots) is the
number of integers including unity less than n and prime to it.
Prove that if ^i, ^s* ^s* ••• ^ the argmnents of the primitive roots, 2coBpS=0 when
p is A positive integer less than -j- — i , where a, b, c^„.k are the different constituent
primes of n; and that, when p=-j- — .> 2co6jp^«» , , , where ii is the number of
the constituent primes.
(Cambridge Mathematical Tripos, Part I. 1895.)
CHAPTER II.
The Theory of Absolute Convergence.
6. The limit of a sequence of quantities.
Let Zi, z^i Zz, ... be a sequence of quantities (real or complex), ififinite in
number. The sequence is said to tend towards a limiting value or limit I,
provided that, corresponding to every positive quantity €, however small, a
number n can be chosen, such that the inequality
I Z^ - Z I < 6
is true for all values of m greater than w. If £: is a variable quantity which
takes in succession the values ^i, ^, ^t> ••• » then z is said to tend to the Ivmit L
Example. Consider the sequence of numbers ^, ^, |)>-<9 for which ^=oii* This
sequence tends to the limiting value 1=0; for if any positive quantity c be taken, and
if n denote the integer next greater than - y^~o > ^^^^ ^^^ inequality
1
is true for all values of m greater than n.
6. The necessary and sufficient condition for the existence of a limit
We shall now shew that the necessary and sufficient condition for the
existence of a limiting value of a sequence of finite numbers Zi, z^, ^,, ... is
that corresponding to any given positive quantity e, however small, it shall be
possible to find a number n such that the equation
is verified for aXl positive integral values of p. This may be expressed in
words by the statement that a finite variable quantity has a limit if and
only if its oscillations have the limit zero ; it may be regarded as one of the
fundamental theorems of analysis.
First, we have to shew that this condition is necessaiy, i.e, that it is
satisfied whenever a limit exists. Suppose then that a limit I exists ; then
5, 6] THE THEORY OF ABSOLUTE CONVERGENCK 9
(§ 5) corresponding to any positive quantity €, however small, a number n
can be chosen such that
and I Zn^ — Z I < ^ , for all values of p ;
therefore
I ^n+p — Zn\<\ (Zn+p — ~ (^n — I
<\Zf^-l\ + \Zn-l\
€ €
which shews the necessity of the condition
I '^«-H> ■" -^n I < ^1
and thus establishes the first half of the theorem.
Secondly, we have to shew that this condition is sufficient, i.e. that if it
is satisfied, then a limit exists. Suppose then that this condition is satis-
fied Let
Zr = a?r + tyr,
where Xr and iyr are the real and imaginary parts of Zr. Then if
I Zf^p — 2r„ I < €,
we have | {x,^ - x^) + i(yn+i> -yn)\<^
and therefore x^ — €< Xn^ < ^n + €,
and yn-€< yn+p < yn + e.
Now the number n is determined by the quantity €, which can be assigned
arbitrarily. Let ni> ^ ^i ^4> ••• be the numbers which correspond in this
€ € € €
way to the quantities ^, j, -, =^,.... Let w* be the least of the quantities
Xn + e, Xn^ + 5, ^n, + 7 , ••• ^n» + o^ , SO that the quantities w©, v^, ti„ ... are a
decreasing sequence ; and let v^ be the greatest of the quantities
€ € €
so that the quantities Vo, Vi, «,,... are an increasing sequence ; and clearly
Then any of the numbers in the li-sequence is greater than any of the
numbers in the v-sequence, since we have
tir>Vr>Vt, if r > «,
and tV > t/, > v„ if r < « ;
^S^K
10 THE PROCESSES OF ANALYSIS. [CHAP. IL
and the difference u^ — v^ can be made as small as we please by increasing
k. These two sequences u and v therefore uniquely define a real number
(rational or irrational) ^, such that { is less than any Dumber in the
iz-sequence and greater than any number in the v-sequence, and the
differences Ut — ^ and ^—vt can be made as small as we please by increasing k.
Then w* - f < w* - v* < ^^^ ,
so k«*-fl<km-ti*|+|t**-f|<2F:Y + 2^<2^-
Moreover, by hypothesis,
where p is any positive integer ; and so
Since -^j—^ can be made as small as we wish by increasing k, this inequality
shews that the sequence a^, aj^, a:,, ... tends to the limit f. Similarly the
sequence ^i, ^at yt> ••• tends to a limit rj.
Thus if T be any small positive quantity, it is possible to choose a number
m such that for all values of r greater than m we have
and therefore (xr - f )* + (yr — f y* < t^,
or \Zr-l\<T,
where Z = f + iff.
This inequality shews that the sequence of quantities Zn z^, Zt, .*. tends to
the limit I ; which establishes the required result, namely that the condition
expressed is sufficient to ensure the existence of a limit.
7. Convergence of cm infinite series.
Let til, t^, t^s, ... tin h^ a series of numbers (real or complex). Let the
sum
be denoted by Sn*
Then the infinite series
is said to be convergent, or to converge to a sum 8, if the sequence of numbers
Si, 82, 89, ... tends to a definite limit 8 aa n tends to infinity. In other
cases, the infinite series is said to be divergent. When the series converges
the quantity 8 — 8n, which is the sum of the series
tifi+i + **fH-« "^ tf n+t + • • • ,
7] THE THEORY OF ABSOLUTE CONVERGENCE. 11
is called the remainder after n terms, and is frequently denoted by the
symbol Rn.
The sum Un+i + Wn+i + ... + thn^
will be denoted by Sn,p-
It follows at once, by combining the above definition with the results
of the last paragraph, that the necessary and sufficient condition for the
convergence of an infinite series is that Sn^p shall tend to the limit zero
as n tends to infinity, whatever p is.
Since tVM = >S^n,i) it follows as a peurticular case that ibn+i must tend to
zero as n tends to infinity, — in other words, the terms of a convergent series
must ultimately become indefinitely small. But this last condition, though
necessary, is not sufficient in itself to ensure the convergence of the series,
as appears from a study of the series
In this series,
or Sn^n>2'
Therefore S = 1 + Si,x 4- iS^., + ^4,4 + fii.« + S^e^ie + ...
^ •■••222 •••>
which is clearly infinite ; the series is therefore divergent.
Infinite series were used by Lord Brouncker in P/dl, Trans, 1668, and the expressions
oonvergent and divergent were introduced by Gregory in the same year. But the great
mathematicians of the 18th century used infinite aeries fredy without, for the most part,
considering the question of their convergence. Thus Euler gave the sum of the series
as zero, on the ground that
...+p + ^ + j+l+«+a«+«»+ (a)
^+^+^+... = -i- (6)
1 ^*
and 14. + 4....= ' (c).
z sr z— 1
The error of course arises from the fact that the series (6) converges only when | ^ | < 1, and the
series (c) converges only when \z\>l,ao the series (a) does not converge for any value otz.
The modem theory of convergence may be said to date from the publication of Gauss'
DisquigitioneM circa seriem infinitam 1+?^+... in 1812, and Cauchy** Analyse Algdhrique
in 1821. See Reifi^ Oeschichte der imendltchen Reihen (Tubingen, 1889).
„ .-■ ^.■" ■ W\. . Jl'.'
12 THE PROCESSES OF ANALYSIS. [CHAP. IL
8. AbsoltUe convergence and semi-convergence.
In order that the series
(which we shall frequently denote by Xun), whose terms are supposed to be
any complex quantities, may be convergent, it is sufficient, but not necessary,
that the series S | z^ | shall be convergent.
For we have
I ^n,P I = I ^Wi + ^n+a + ... + ^n+p \
< I l^n+i I + I ^+9 I + ... 4- I Un+p I ,
and this last expression is inBnitely small, whatever p may be, when n is
infinitely great, provided the series S 1 2^ | is convergent.
Although this condition is sufficient to ensure the convergence of the
series Xun, it is not necessary, i.e. the series Sun can converge even when
the series 2 1 Wn | diverges. This may be seen by considering the series
1 2 + 8 4+6 ••• + n +••••
This series is convergent ; for writing it in the form
1.1.1.
or 2 12 + 30+*"'
we see that its sum is greater than ^ > And that the partial sum obtained by
truncating the series after its 2nth term increases as n increases; on the
other hand, by writing it in the form
we see that the sum is less than 1, and that the partial sum obtained by
truncating the series after its (2n + l)th term decreases as n increases.
These partial sums must therefore tend to some limit between x and 1, and
so the series converges. But the series of moduli is
^2^3^4+'*''
which as already shewn is divergent. In this case therefore, the divergence
of the series of moduli does not entail the divergence of the series itself.
Series whose convergence is due to the convergence of the series formed
by the moduli of their terms possess special properties of great importance,
and are called absolutely convergent series. Series which though convergent
are not absolutely convergent (i.e. the series themselves converge, but the
series of moduli diverge) are said to be semi-convergent or conditionaUy
convergent
8, 9] THE THEORY OF ABSOLUTE CONVERGENCE. 18
CO 1
9. The geometrical series, and the series S — .
The convergence of a particular series is in most cases investigated, not
by the direct consideration of the sum Snp, but (as will appear from the
following articles) by a comparison of the given series with some other series
which is known to be convergent or divergent. We shall now investigate
the convergence of two of the series which are most frequently used as
standards for comparison.
(1) The geometrical series.
The geometrical series is defined to be the series
l + irH--j' + ^ + 2^....
Considering the series of moduli
we have for it iSfn.i,= kl'^' + l-^^l^+'H- ... +|«^|*^,
or ^n.p=l^r^j^.
Now if I -2: 1 < 1, then -= — r-K- is finite for all values of jp, while \z\^^^ tends
1-1*.
to zero as n tends to infinity. The series
1+ U| + U|>+...
is therefore convergent so long as 1 2: { < 1, and therefore the geometric series is
ahsolutely convergent so long as \z\<l.
When I ir I > 1, the terms of the geometric series do not tend to zero as n
increases, and the series is therefore divergent.
(2) 2%««ertC8 j;; + 2j + gi+^+g, + ....
« 1
Consider now the series 2 — , where s is any positive real quantity.
We have 2^ + 3^ < 2"*'^ 2^^ '
11114 1
4« "^ 5* "^ 6* "^ 7« ^ 4' "^ 4^1 '
and so on. Thus the sum of any number of terms of the series is less than
the sum of the corresponding terms of the series
1 _1 11
l»-i "^ 2«-^ "*" 4«"* S*-^ '
1 J_ 1 1
^^ p=5"*" 2«-i "^ 2*^"*> ■*"2»(»-i) ■^•••»
14 THE PROCESSES OF ANALYSIS. [OHAP. II.
and hence the convergence of this last series would involve that of the
original series. But this last series is a geometrical series, and is therefore
convergent if
1 ,
that is, if 8> 1.
The series ^ —i is therefore convergent if s>l; and since its terms
n=l w*
are all real and positive, they are equal to their own moduli, and so the series
of moduli of the terms is convergent ; that is, the convergence is absolute.
If 5 = 1, the series becomes
which we have already shewn to be divergent; and when «»!, it is d fortiori
divergent, since the etfect of diminishing « is to increase the terms of the
• 1 .
series. The series 2 — w therefore divergent if s^l.
10. The Comparison-Theorem.
We shall now shew that a series
»
wUl be absolutely convergent^ provided 1 1^ | w always less than C\vn\, where
C is any finite number independent of n, and v^ is the nth term of another
series which is known to be absolutely convergent.
For we have under these conditions
I Un+i i + I Un+t I + . . . + I Un+p I < C^ 1 1 ^n+i | + | Vn+3 I + • • • + | t^n-hp I } >
where n and p are any integers. But since the series Svn is absolutely
convergent, the series 2 | Vn | is convergent, and so
tends to zero as n increases, whatever p may be. It follows therefore that
tends to zero as n increases, whatever p may be, i.e. the series 2 | Wn | is
convergent. The series 2t*n is therefore absolutely convergent.
Corollary, A series will be absolutely convergent if the ratio of its
terms, to the corresponding terms of a series which is known to be abso-
lutely convergent, is always finite.
Example 1. Shew that the series
cos « + Si cos 2«H-si cos 3« + 7% COB 4« + . ..
2' o' 4'
is absolutely convergent for all real values of t.
10] THE THEORY OF ABSOLUTE CONVERGENCE. 15
For wheD z ia real, we have | cobtl? | ^ 1, and therefore | ^g i ^ "§ • Th® moduli of
the terms of the given series are therefore less than, or at most equal to, the corresponding
terms of the series
1 11
which by § 9 is absolutely conveigent The given series is therefore absolutely convergent.
Example 2. Shew that the series
1 1
V . ••>
where z^^(^l+^e^, (n=l, 2, 3, ...)
is conveigent for all values of z^ except the values z^z^ z^^z^,*...
The geometric representation of comjdez numbers is helpful in discussing a question of
this kind. Let values of the complex number z be represented on a plane : then the values
'i) Hi %)••• ^U ^*^i^™ & series of points which for large values of n lie very near the
circumference of the circle whose centre is the origin and whose radius is unity: so
that in &ct the whole circumference of this circle may be r^arded as composed of points
included in the values z^^.
For these special values z^ of r, the given series is clearly divergent, since the term
becomes infinite when z=z,^. The series is therefore divergent at ^U points z
situated on the circumference of the circle of radius unity.
Suppose now that z has a value which is distinct from any of the values z^. Then
is finite for all values of n, and less than some definite upper limit c : so the moduli
of the terms of the given series are less than the corresponding terms of the series
which is known to be absolutely convergent. The given series is therefore absolutely
convergent for all values of r, except the values z^.
It is interesting to notice that the area in the 2-plane over which the series converges
is divided into two parts, between which there is no intercommunication, by the circle
1*1-1.
Example 3. Shew that the series
2sin^+4sing+8sin — + ... + 2*sin ^-h.--
converges absolutely for all finite values of z.
For when n is large, the quantity
2,„ • WW
"sm-
2«
3*
has a value nearly unity; the given series is therefore absolutely convergent, since the
comparison series 2 * ' is absolutely convergent.
16 THE PROCESSES OF ANALYSIS. [CHAP. II.
11. Disctission of a special aeries of importance.
The theorem of § 10 enables us to establish the absolute convergence
of a series which will be found to be of great importance in the theory of
Elliptic Functions.
Let Q>i and o), be any constants whose ratio is not purely real; and
consider the series
Us
z^
>r
{z — 2ma)i - 2nQ),)* (2ma>i + 2nft)jy
where the summation extends over all positive and negative integral and
zero values of m and n (the simultaneous zero values m = 0, n = excepted).
At each of the points z = 2ma>i + 2nQ>s one term of the series is infinite, and
the series therefore is not convergent. The absolute convergence of the
series for all other values of z can be established as follows.
Let z have any value not included in this set of exceptional values.
The series may be written
L + s ^
^ (2ma>i + 2n(o^
1-
r-^-
2mQ>i + 2nci>,>
Now when | 2ma>i + ^na>^ \ is large (and we can suppose the series arranged
in order of magnitude of | 2m6)i + 2m»2|), we have
1 1 f r_i
Limit o = ■!•
2m(it)i + 27ica2
The series is therefore absolutely convergent if the series
2 ^
(2ma>i -♦- 271012)*
is absolutely convergent : that is, if the series
2 ^ -
(2wicoi -f- 2r2G)a)'
is absolutely convergent.
To discuss the convergence of the latter series, let
cDi = ai + t^Si , ©a = Oa + iySj,
where aj, Oa, ySi, ^Sj, are real. Then the series of moduli of the terms of this
series is
This converges if the series
2 •- (which we may denote by S)
(m* 4- n^y
■i
11, 12] THE THEORY OF ABSOLUTE CONVERGENCE. IT
converges ; for the quotient of corresponding terms is
where M = - ;
and this is never zero or infinite.
We have therefore only to study the convergence of the series S. Now
iit= -00 n= - 00 (m* + n^y
00 00 1
= 4 2 2
where in the summation the occurrence of the pair of values m = 0, n =
together is excluded.
Separating S into the terms for which m = n, m>n, and m<n, re-
spectively, we have
00 1 00 yn-1 1 CO »-l 1
i/Sf=2 -4-i+2 2 --^ — i+2 2
m=i(2m')* w=i n=o (m'4-n*)* »-i »t=o (w' + ?i*)*
»»-! 1 Wl 1
But 2 . < i < — ;, .
Therefore jfif< | -_^+ f 1+ i 1
t=i 2*m' «=i ^ «=i ^
00 I ** 1
But the series 2 —1 and 2 -„ are known to be convergent. So the
series S is absolutely convergent. The original series is therefore absolutely
convergent for all values of z except the specified excluded values.
Example, Prove that the series
1
2
{m^-\-m^-\- . . . 4-wir^r
in which the summation extends over all positive and n^ative integral values and zero
values of m,, m,, ... m,, except the set of simultaneous zero values, is absolutely convergent
if yi>- . (Eisenstein, CrelUs Journal^ xxxv.)
It
12. A convergency-test which depends on the ratio of the successive terms
of a series.
We shall now shew that a series
^1 + ^ + Us + ^4 + • • •
is absolutely convergent, provided that for all valves of n greater than some
w.A. 2
18 THE PROCESSES OF ANALYSIS. [CHAP. II.
fixed value r, the quantity \ — ^1 is less than K, where K is some positive quantity
Un
independent of n and less than unity.
For the terms of the series
are respectively less than the terms of the series
which is a geometric series, and therefore absolutely convergent when K <1.
Thus if —^ tends as n increases to a limiting value which is less than
unity, the series is absolutely convergent.
Example 1. If | c |< 1, shew that the series
11=1
converges absolutely for all values of z.
For the ratio of the (n + l)th term to the nth is
or c**+V,
and if I c |<I, this is ultimately indefinitely smalL
Example 2. Shew that the series
,.«"fe^, (a-6)(a-2fe) (a-fe)(a-26)(a-36)
converges absolutely so long as 1 2 |<-T-i .
For the ratio of the (»+l)th term to the nth is — r ^> ^^ ultimately - 6z : so the con-
dition for absolute convei^ence is 1 62; |<1, or U|<--t-: .
Example 3. Shew that the series 2 converges absolutely so long as
2|<1.
For when |2|<1, the terms of the series bear a finite ratio to those of the series
2 n2"~^; but th^ latter series is then absolutely convergent, since the ratio of the
00
(n+ l)th term to the nth is f 1 H — j z^ which tends to a limit less than unity as n increasea
13. A general theorem on series /or which Limit
»=oo
^n+i
= 1.
It is obvious that if, for all values of n greater than some fixed value r.
13]
THE THEOBY OF ABSOLUTE CONVERGENCE.
19
t^n+i I is greater than \iLn\, then the terms of the series do not tend to zero as
w«
n+i
U
n
n increases, and the series is therefore divergent. On the other hand, if
is always leas than some quantity which is itself less than unity, we have
shewn in § 12 that the series is absolutely convergent. The limiting case
is that in which, as n increases,
u
n
tends to the value unity. In this case
a further investigation is necessary.
We shall now shew that a series
Wi + W2 + t^+ '",
in which
^+1
tends to the limit unity as n increases, will he absolutely con-
m
vergent if, for ail values of n after some fixed value, we have
<1-
1 + c
n
where c is a positive quantity independent of n.
For compare the series 2 | t/n | with the convergent series Svn* where
A
v«= -
n
i+,
and -4 is a constant ; we have
Vn Vn+l/ \ nJ
-(-I)
1+1 .11
= 1 — -^— -I- terms in — , — ,
n n^ if
V,
As n increases, -""^^ will therefore tend to the limit
Vn
1-
Hi
n
so that after some value of n we shall have
^+1
Wn
Vn+i
V
n
By a suitable choice of the constant A, we can therefore secure that for
all values of n we shall have
As Sv^ is convergent, 2 1 1^ | is therefore convergent, and so 2wn is abso-
lutely convergent.
2—2
20
THE PROCESSES OF ANALYSIS.
[chap. n.
Corollary. If
form
^+1
^
can be expanded in descending powers of n in the
where -4i, A^, -4,, ... are independent of n, then the series is absolutely
convergent if -4i < — 1.
This is easily seen to follow from the fact that when n is large the terms
become unimportant in comparison with A^,
14. Convergence of the hypergeometric series.
The theorems which have been given may be illustrated by a discussion
of the convergence of the hypergeometric series,
a,b a(a + l)b(b+l)
■^l.c "^ 1.2.c(c + l)
a(a-H)(a-f 2)6(6-H)(6 + 2)
1.2.3.c(c + l)(c+2) ■*■•••'
which is generally denoted by F (a, 6, c, z).
If c is a negative integer, all the terms after the (1 — c)th will be infinite ;
and if either a or 6 is a negative integer the series will terminate at the
(1 — a)th or (1 — 6)th term as the case may be. We shall suppose these
cases set aside, so that a, 6, and c are assumed not to be negative integers.
The ratio of the (n -♦- l)th term to the nth is
Un+i (a -♦- M — 1) (6 4- n - 1)
= z.
Therefore
Un
^n+i
w(c+w — 1)
U.
n
1 +
a- 1
n
1 +
6-1
n
1 +
c-1
n
As n tends to infinity, this tends to the limit \z\. We see therefore by § 12
that the series is absolutely convergent when \z\<l, and divergent when
\z\>h
When 1^1=1, we have
^+1
1 +
g-l
n
1 +
6-1
n
l_£^^(- !)•_...
n
n"
1 + - ^ -+ terms m -- , — , etc.
n n^ n'
Now a, 6, c are in the most general case supposed to be complex numbers.
14, 15] THE THEORY OF ABSOLUTE CONVERGENCE. 21
Let them be given in terms of their real and imaginary parts by the
equations
a = a' 4- ia!\
c = c' -f ic".
Then (neglecting the terms in — , — , etc.) we have
1% Iv
^+1
ajH6W;-H-i(a" + 6"-c")
n
= 1 +
a' + h'-c'-W fa" 4 h" - c"\») *
n
) - (" ' „
a' + 6'-c'-l , .1 1 ^
= 1 H h terms m — , — , etc.
By § 13, the condition for absolute convergence is
a' + 6'~c'<0.
Hence when \z\=^\,ihe condition for ike absolute convergence of the hyper-
geometric aeries is that the real part of a + b — c shall be negative,
16. Effect of changing the order of the terms in a series.
In an ordinary sum the order of the terms is of no importance, and can
be varied without affecting^the result of the addition. In an infinite series
however this is no longer the case, as will appear from the following example.
T^f 'C 1.1 1.1.1 1.1.1 1.
and Sf=i -1+1-1+1-1 +
and let 2n and Sn denote the sums of their first n terms. These infinite
series are formed of the same terms, but the order of the terms is different.
Then if A: be any positive integer,
n * 11111
But ^*-^*-' = 2/fc— l+2A-]k=2^T-2ifc-
Similarly p,_, -p,_ = ^ - -^-_-^ .
A series of equations like this can be formed, of which the last is
Adding these, we have
,111 1 c»
22 THE PROCESSES OF ANALYSia [CHAP. II.
Thus 2jt = So: + s Safc.
2
Making k indefinitely great, this gives
an equation which shews that the eflfect of deranging the order of the terms
in S has been an alteration in the value of its sum.
Example, If in the series
the order of the terms be altered, so that the ratio of the number of positive terms to the
number of negative terms in S^ is ultimately a\ shew that the sum of the series will
become log (2a).
(Manning.)
16. The fundamental property of absolutely convergent aeries.
We shall now shew that the sum of an absolutely convergent series is not
affected by changing in any manner the order in which the terms occur
For let iS = t^i + 1^, 4- 1/^ + ^4 + . . .
be an absolutely convergent series, and let /S' be a series formed by the same
terms in a different order.
Suppose that in order to include the first n terms of Sy it is necessary to
take m terms of S\ So if k be any number greater than m, we have
^k = 'Sin + terms of S whose suflBx is greater than n.
Therefore
I 'S'A;' — /S I < I /Sin — /Si j -♦- the sum of the moduli of a number of terms of S
whose suffix is greater than n
When n tends to infinity, | /?„ — ^ I tends to zero since the series 8 is con-
vergent, and the sum
tends to zero also, since the series is absolutely convergent.
Thus I St' — /Si I tends to zero when k is indefinitely increased; which
establishes the required result.
17. Riemann*s theorem on semi-convergent series.
We shall now shew that a semi-convergent series
Wl + ^ + 1^ + ^4 + . . . ,
with real terms, may be made to converge to any desired real value, by suitably
disposing ike order in which the terms occur. This property stands in sharp
contradiction to that proved in the last article; an example of it was
afforded by the result of §15.
16, 17] THE THEORY OF ABSOLUTE CONVERGENCE. 28
To establish the theorem, let the positive terms in the series be
"^j '^> ^i>,i ••• >
and let the negative terms be
'^j*,* "^1 "^i*,! ••• •
Then the series
and -^i»,-^-^ii,- ...
cannot be both convergent : for if they were, the original series would be
absolutely convergent: one of them must therefore be divergent: and the
other cannot be convergent, since in that case the original series would be
divergent. It follows that the series
tAp, + 1^, + tij^ + . . .
and — t^, — ^n,-^— ...
are both divergent.
Now let S be any real number, and let it be desired to change the order
of the terms in the original series, in such a way as to cause it to converge
to the sum &. Suppose that a terms of the series
have to be taken in order to obtain a sum greater than fif, so that
Take now a number 6 of the terms of the series
such as are required to make the sum
less than & : so that
Take next a number c of the terms of the series
such as are required to make the sum
greater than S ; and then take a number d of the terms of the series
in such a way as to make the sum
less than & again ; and so on.
Proceeding in this way, we obtain a series whose sum at any stage of
24 THE PROCESSES OF ANALYSIS. [OHAP. II.
the process, differs from S by less than the last term included. But the
terms of the series
lii + ^,4- w,+ ...
are ultimately indefinitely small, since the series is convergent; we can
therefore in this way obtain a series
whose sum differs from S by as little as we please ; and it consists of the
terms of the original series, disposed in a different order. This establishes
the result above stated.
Corollary. If the terms of the original series are complex, they can be
disposed in such an order as to give an arbitrarily assigned value to either
the real or the imaginary part of the sum.
18. Cauchys theorem on the multiplication of absolutely convergent series.
We shall now shew that if two series
and T= Vi + Vj + Vt 4- ...
are absolutely convergent, then the series
formed by the products of their termSy written in a/ay order, is ahsohitely con-
vergenty and has for sum ST.
Suppose that in order to include all the terms of the product
it is necessary to take m terms of P ; and let k be any number greater
than m.
Then
-P* = (^ + ^+ ... +^n)(Vi + ^2+ ••• +Vn) + terms t^aV/i in which either a or yS
is greater than n,
so I Pjt- Sri < I Sr,Tn'-ST\ 4- terms | w. 1 1 v^ |.
Let {u-k-p) be the greatest suffix contained in these suffixes a and /8.
Then
|P,-OT:^|Snrn-«2'| + lhf„+,|4-...4-|^n+pll{iVi;4-...4-|t;n+p|}
4- {i Wi I 4- ... 4- I Mn 1} (i V„+, I 4- ... 4- I Vn+p I}.
Now when n tends to infinity,
I ^n+i i 4- 1 Un+% j 4- ... 4- 1 Un^p \ tcuds to zero,
and I Vn+i I 4- ... 4- 1 Vn^p \ tends to zero,
while their coefficients tend to finite limits.
Therefore | P^ — ST \ tends to zero, which proves the theorem.
18, 19] THE THEORY OF ABSOLUTE CONVERGENCE. 25
Example 1. Shew that the series obtained by multiplying the two seriee
l+£ + ^+f' + ?! +
2 2* 2' 2* '''
and l+- + -2+'5+ — >
Z Z /u
converges so long as the representative point of z lies in the ring-shaped r^on bounded
by the circles l^jal and |z|a2.
For the first series converges only when |«|<2, and the second only when |«|>1, and
both must converge if the product is to converge.
Example 2. Prove by multiplication of series that
{cos S? cos 52 1 fTT* 2 /cos 2^ cos 4? . \) . cos a? . cos 5^ .
For the coefficient of cos (2r-|-l) 2 in the product on the left-hand side of the equation is
_jr« 1 ; 1 f 1 , 1 1
9 (2r+l)« 3 *=, (Uf \{U - 2r- 1)« "^ (2it+2r4- 1)«J '
or
or
or
or
or
9(2r+l)2 3(2r+l)«tli\V2it-2r-l 2/?/ "*" W 2Xr+2r+l/ J'
^2 1 • r 2 2 4 11
r_2 2 4 1
t(2i-)2'*"(2it-l)2 (2;fc-2r-l)(2ir4-2r+l)J
9(2r+l)« 3(2r+l)«fcfi l(2i-)2^(2it-l)2 (2;fc-2r- l)(2ir4-2r+l)J ^3(2r4-l)*'
9(2r + l)2'*"3(2r-hl)* 3(2r+l)2V 22'^3«'^42'^"7 '*"3(2r+l)*'
»r2 . 1 2 TfS
9(2r+l)2^(2r+l)* 3(2r+l)« ' 6 '
1
(2r+l)*'
which gives the required result.
19. Mertens' theorem on the multiplication of a semi-convergent series hy
an absolutely convergent series.
We shall now shew that if a series
S = i^-fi^ + w, + ...
is semi-convergent, and another series
is absolutely convergent, then the series
where pn = U^Vn + t^Vn-i + ... + UnV^,
is convergent, and its sum is ST.
For Pn = the sum of all terms UaVp in which a + /S < n + 1
=^(Ui + 1U+,,.+Un)(Vi-{-V2+..,+Vn)-VjUn-Vi{Un'¥Unr^i)-...
26 THB PROCESSES OF ANALYSIS. [CHAP. II.
Therefore
|Pn-/Sfri<|/Sf„r„-/gr| + |w„||t;,| + |t;,|K+un-,l + ...
Now let k denote some number about half-way between 1 and n\ let € be
the greatest of the quantities
and let 7 be the greatest of the quantities
i ^n + . . . + Wn-*-i 1 1 • • • ! 1^ + ^n-i + . . . + ti<| | .
Then
As n tends to infinity, € and { | y^^., | + . . . + 1 Vn 1 } are infinitesimal, while
{|i^s| + ... +1 Vfc+a|} and 7 are finite. So every term on the right-hand side
of the last equation is infinitesimal, and therefore in the limit
P = /gT,
which establishes the theorem.
20. AbeVa resuU on th^ multiplication of aeries.
We shall next prove a still more general theorem due to Abel*, which
may be stated thus :
Let two series 2 u^ and 2 Vn converge to the limits U and V respec-
tively, and let the quantity
be denoted by Wn. Then if the series
converges at all, it converges to the sum UV,
It will be noticed that none of the series considered need be absolutely
convergent.
We shall follow a method of proof due to Cesarof.
Lemma I. If a set of quantities «i, 52> *8> ••• t^^ ^ ^ limit s, then
. . 1 *
Limit - 2 «i = «.
f»=oo n ,=1
For if € be any small positive number, we can find a number k such that
the inequality
\Sr — s\<€
is satisfied for all values of r greater than k. We have therefore
2 n 1 * 1 *"** 1 *
- 2 «< = - 2 «i + - 2 « + - 2 («f — 5).
Ui^i ni=i n i^jc ni^jc
• Crelle'B Journal, i. (1827).
t Bulletin des Sciences math. (2) xiv. (1890).
20] THE THEORY OP ABSOLUTE CX)NVERGENCE. 27
Thus - 2 «i s . < - 2 ^< + - 2 5i - 5 1
1 4 1 I n-k + l
< - 2 \8i\-\ €.
TV t==l W
1 *
Now make n infinitely great compared with k ; then - 2 \si\ tends to zero,
^ f=i
fl Jfc 4- 1
and tends to unity,
1 \
Limit - 2 «t- - fi < € ;
and so
and as e can be made as small as we please, this establishes the Lemma.
Lemma II, If, as n increases indefinitely^ On and bn tend respectively
to the limits a and b, then
Limit -{aibn + a^bn-i + . . . -h Onfti) = ab.
n=oo n
To prove this, let v be the greatest integer contained in ^n. Then if e be
any small positive number, we can take n so great that the inequality
6^ - 6 I < 6
holds so long as r > n-^p.
Hence |ai(6n-6) + aa(6^i- 6) + ... + a^(6n-r+i — b)\
< €{|ai| + I Oal + ... + I a„|}.
Hence Limit - \ Oi (bn— b) + a^(bn^i — b) + ... + a„(6,^_„4.i — 6) |
n
n=cc
<€Limit-{|ai| + | 09! + ... + |a„|}
91=00 n
< € I a I , by Lemma I.
The right-hand side of this inequality can be made as small as we
please; hence
Limit - {oi (bn — 6) + a, (b^^i — 6) + ... + a„ (^n-i^+i — b)] = 0,
n=<» n
or Limit - (ai6„ + a^bn-j -!-...+ a„6»«„+i)
n=oo W
= ^6 X Limit- (oi + a? + . . . + a^)
y=oo ^
= ^a6, by Lemma I.
Similarly
Limit - (a^+ibn-y + a^ibn-v+j + ... 4- (tnbi) = JaJ.
n=« ^
28 THE PROCESSES OF ANALYSIS. [CHAP. II.
Adding the last two equations, we have
Limit (Oibn + Ojin-i + . . • + Onbi) = oi,
11=00
which establishes Lemma II.
Now let Wn denote the sum of the n first terms of the series
considered in the above enunciation of Abel's result, we have
where Un and F„ are used to denote the sums of the first n terms of the series
U and V, From thiswe have
and so by Lemma II. it follows that
Limit -(Tri-fTrj+ ... + Trn)= ^F.
1»=ao W
But if the set of quantities TTj, TFj, ]F,, ... tend to a limit W, we have
by Lemma I.
Limit - ( TT, + TT, + . . . + F„) = W.
Hence W=UV,
which establishes Abel's result.
Example 1. Shew that the series
1-2+2-1 +
is convergent, but that its square (formed by AbePs rule),
2
V2
is divergent.
■--Hfa-S-a-^)-"
Example 2. If the convergent series
0—1 2^ y 4r"^'*'
be multiplied by itself, the terms of the product being arranged as iu AbePs result, shew
that the resulting series is divergent if r ^ ^, but that it converges to the sum ^S^ when
r<i.
(Cauchy and Cajori.)
21. Power-Series,
A series of the type
in which the quantities Oo, ch, a,, a, ... are independent of z, is called a series
proceeding according to ascending powers of z, or briefly a power-series.
21]
THE THEORY OF ABSOLUTE CONVERGENCE.
29
We shall now shew that if a power-series converges for any value Zq of z,
it will he absolutely convergent for all values of z whose representative points
are within a cirde^ which passes through z^ and has its centre at the origin.
00
For if z be such a point, we have | -? | < | «o !• Now since ^a^zj^ converges,
the quantity a^z^ must tend to zero as n increases indefinitely, and so
we can write
Jn
anl =
^p • • • •
where €» tends to zero as n increases. Thus
Ittol + loil |^| + |a«|U(^ + ... = €o + €i -
Zy
Now ultimately every term in the series on the right-hand side is less
than the corresponding term in the convergent geometric series
z
z
s
Z
+ e.
—
+ €,
Zq
Zq
^0
00
n=0
Z
^0
n
the series is therefore convergent; and so the power-series is absolutely
convergent, as the series of moduli of its terms is a convergent series ;
which establishes the result stated.
It follows from this that the area in the ^-plane over which a power-
series converges must always be a circle ; for if the series converges for any
point outside the particular circle which has just been found, we can (by
taking this point as the point Zq) obtain a new and larger circle within which
the series will converge.
The circle in the ^r-plane which includes all the values of z for which
the power-series
ao + ttiZ + a22^ + a^z^ + ...
converges, is called the circle of convergence of the series. The radius of
the circle is called the radius of convergence.
The radius of convergence of a power-series may be infinitely great;
as happens for instance in the case of the series
which represents the function sin z ; in this case the series converges for all
finite values of z real or complex, ie. over the whole 2:-plane.
On the other hand, the radius of convergence of a power-series may be
infinitely small ; thus in the case of the series
1 -h 1! ^ + 2! £:«-h 3! -2^ + 4! ^ + ...,
we have
u
n-fi
^n\z\.
30 THE PROCESSES OF ANALYSIS. [CHAP. II.
which, for all values of n after some fixed value, is greater than unity when
z has any value different from zero. The series converges therefore only at
the point -? = 0, and its circle of convergence is infinitely small.
A power-series may or may not converge for points which are actually on
the circumference of the circle ; thus the series
z z^ 2? z*
whose radius of convergence is unity, converges or diverges at the point 2: = 1
according as « is greater or not greater than unity, as was seen in § 9.
22. Convergency of series derived from a power-series.
Let aQ + aiZ + a^z^ + a^z^-^- a4Z* + ...
be a power-series, and consider the series
Oi -{• 2aiZ -\' SotZ* + ia^z* -^ ...,
which is obtained by differentiating the power-series term by term. We
shall now shew that the derived series has the same circle of convergence as the
original series. .
For let -? be a point within the circle of convergence of the power -series ;
and choose a positive quantity r, intermediate in value between | z \ and the
radius of convergence. Then, since the series 2 an r^ converges absolutely, its
n=0
terms must decrease indefinitely as n increases; and it must therefore be
possible to find a positive quantity M, independent of n, such that the
inequality
M
is true for all values of n.
Then the terms of the series
in\an\\z\^-'
are less than the corresponding terms of the series
But in this series we have
Un n r \ nj r '
which, for all values of n greater than some fixed value, is constantly less than
unity ; this comparison-series therefore converges, and so the series
in\an\\z\^'
22, 23] THE THEORY OF ABSOLUTE CONVERGENCE. 31
converges ; that is, the series 2 non^^^ converges absolutely for all points z
«— 1
00
situated within the circle of convergence of the original series 2 On-s^, and the
two series have the same circle of convergence.
Similarly it can be shewn that the series 2 -^^^ , which is obtained by
integrating the original power-series term by term, has the same circle of
00
convergence as 2 OnZ^K
23. Infinite ProdiLcts.
We proceed now to the consideration of another class of analytical ex-
pressions, known as infinite products.
Let l+Oi, l + Oj, l+Os, ... be an infinite set of quantities. If as
n increases indefinitely, the product
(H-ai)(l + aj)(l + a,)...(l +an)
(which we may denote by 11^) tends to a definite limit other than zero, this
is called the value of the infinite product
n = (l+ai)(l+a3)(l +0,) ...,
and the product is said to be convergent
The product is often written n (1 -f- a«).
If the value of the product is independent of the order in which the
factors occur, the convergence of the product is said to be absolute.
The condition for absolute convergence is given by the following theorem :
in order that the infinite product
(l-f-ai)(l+a,)(l + a,)...
may he ahsolutely convei^gent, it is necessary and sufficient that the series
' Oi -f- Oj + aj -h . . .
should he ahsolutely convergent.
For Iln = e^**'^^^'''^"*"***^"*'*^"^-'*"^**^^"*'*'^,
so that n is absolutely convergent or not according as the series
log(l + Oi) 4- log (1 + Og) -h log (1+ a,) + ...
is absolutely convergent or not. But since log(l + o^) is nearly equal to ar
when Or is small, the terms of this series always bear finite ratios to the
corresponding terms of the series
and so the absolute convergence of one series entails that of the other ; which
establishes the result*.
* A disooBsion of the convergence of infinite products, in which the results are derived
withont making use of the logarithmic function, is given by Pringsheim, Math, Ann. xxxoi.
pp. 119—164.
^
32 THE PROCESSES OF ANALYSIS. [CHAP. II.
ExcMiple, Shew that the infinite product
sin z sin \z sin ^z sin \z
z ' ^z ' '~^z ' iz
is absolutely convergent for all values of z,
. z
sm-
For when n is large, ^~~ is of the form 1 — ^ , where \^ is finite ; and the series
n
* X * 1
2 -^ is absolutely convergent, as is seen on comparing it with 2 ,« . The infinite pro-
11=1 n* n=i w
duct is therefore absolutely convergent.
24. Some examples of injmite products.
Consider the infinite product
m
sm 21
which represents the function .
In order to find whether it is absolutely convergent, we must consider the
* ^2 ^* * 1
series 2 ^-r , or — 2 — ; this series is absolutely convergent, and so the
product is absolutely convergent for all finite values of z.
But now let this product be written in the form
(-i)(>-J)('-i)('-4)--
The absolute convergence of this product depends on that of the series
z z z z
But this series is only semi-convergent, since its series of moduli
z\ \z\ \z\ \z'
+ — -!--- +'77-'+...
7r TT 27r 27r
is divergent. In this form therefore the infinite product is not absolutely
convergent, i.e. if the order of the factors ( 1 ± - 1 is deranged there is
a risk of altering the value of the product.
Lastly, let the same product be written in the form
M)'-H('n)«1l('-l^)«"}{('-a-
in which each of the expressions
1 + I e mir
>■
24] THE THEORY OF ABSOLUTE CONVERGENCE. 33
is counted as a single term of the infinite product. The absolute convergence
of this product depends on that of the series
or
( 27r»'*'"V'^( 27r^'^-)"^r27r«.2«'^-'V"^(~27r^:T^'*'--j'
and the absolute convergence of this series follows from that of the series
The infinite product in this last form is therefore again absolutely
convergent, the adjunction of the factors e *'" having changed the con-
vergence from conditional to absolute.
Example 1. Prove that n -[(1 ) «*[ is absolutely convergent for all values of
Zj if c is a constant other than a negative integer.
For the infinite product is absolutely convergent provided the series
Le. if 2 < — ii^—+ terms m — _, —. etcV is,
* 1
and- on comparison with the convergent series 2 —^ , this is seen to be the case.
Example 2. Shew that n jl — (l — ) «"*[ converges for all points z situated
outside a circle whose centre is the origin and radius unity.
For the infinite product is absolutely convergent provided the series
n=2 \ nj
is absolutely convergent. But as n increases, (1 — ) tends to the finite limit e, so the
ratio of the (n + l)th term of the series to the nth term is ultimately - ; there is therefore
z
absolute convergence when , -
<1, or |z|>l.
Example 3. Shew that
1.2.3...(w-l)
— - -> - — ^- — -- n'
z{z+l)iz + 2).„(z-\-n-\)
tends to a finite limit as n increases indefinitely, unless « is a negative integer.
W. A. 3
34 THE PROCESSES OF ANALYSIS. [CHAP. II.
For the expression can be regarded as a product of which the nth term is
This product is therefore absolutely convergent, provided the series
is absolutely convergent ; and a comparison with the convergent series 2 —^ shews that
this is the case. When 2 is a negative integer the expression clearly becomes infinite owing
to the vanishing of one of the factors in the denominator.
Example 4. Prove that
'(■-^)(-i)(-:)('-s)(>-4i)('-i>-'-'--^^
For the given product
-■ir:"(-i)(-i)('-;)-{'-<M^)('-K)('-s)
*\ ' 2^ 8 4^1 • 2fc-l 2*^*/
:= Limit
xz
(>-;)- ('-s)'*-('-4)-^- (■-«)'-
=.Uimte'H^'lH-"'^2hi-k)z(l--^e'(l-{-^e'i(^
since the product whose factors are
is ahsolviely convergent and so the order of its factors can be altered.
Since log2 = l -Hi-Hi- - >
this sliews that the given product is equal to
e V wn.z.
26. Cauchys theorem on products which are not absolutely convergent
We shall now shew that if
ai + aa + cis + a4+ ...
is a semi-convergent sei-ies of real terms, then the infinite product
(l+OiXl+OaXl + Oj)...
26, 26] THE THEORY OF ABSOLUTE CONVERGENCE. 36
converges (though not absolutely) or diverges {to the value zero), according
as the series
ai*+ aj'H- 03*+...
is convergent or divergent.
For the infinite product in question converges (though not absolutely)
or diverges (to the value zero) according as the series
log (1 + a,) + log (1 + Oa) + ...
is semi-convergent or diverges to the value — 00 .
n««oo
Now since the series 2 On is convergent, the quantities On ultimately
«=i
diminish indefinitely, and therefore we can write
a,»
log (1 + On) = a„ - ^ (1 + €n),
where |€n| tends to zero as n tends to infinity.
»=•
If the series 2 a^* diverges, it is clear therefore that the series 2 log(l + a^)
n=l
must diverge to the value — 00 ; if on the other hand the series 2 an* con-
n=l
n=sao
verges, the series 2 log (1 + On) is convergent. From this the results relating
n=l
to the infinite product follow at once.
26. Infinite Determinants.
Infinite series and infinite products are not by any means the only known
cases of infinite processes which can lead to convergent results. The re-
searches of Mr G. W. Hill in the Lunar Theory* brought into notice the
possibilities of infinite determinants.
The actual investigation of the convergence is due not to Hill but to Poincare, Bull, de
la Soc. MoUk. de France^ xiv. (1886), p. 87. We shall follow the exposition given by
H. von Koch, Ada Math. xvi. (1892), p. 217.
Let Aije (i. A: = — X , . . . + 00 ) be a doubly-infinite set of given numbers, and
denote by
the determinant formed of the quantities ili^ (i, A = — m — . . . + m) ; then if,
for indefinitely increasing values of m. the quantity D^ has a determinate
limit Z), we shall say that the infinite determinant
is convergent and has a value D, In the case in which the limit D does not
exist, the determinant in question will be said to be divergent.
* Beprinted in Acta Maihematicay rm. pp. 1 — 86 (1886).
3—2
36
THE PROCESSES OF ANALYSIS.
[chap. II.
The elements Au{i = — <x> ,,. -^ ao) are said to form the prindpcU diagonal
of the determinant D ; the elements Aijc{k = oo .,, + oo) are said to form the
line i ; and the elements -4ijt(t = — oo...-f-x) are said to form the column k.
Any element A^ is called a diagonal or a non-diagonal element, according
Sisi = k or i^ k. The element Aq^q is called the origin of the determinant.
27. Convergence of an infinite determinant
We shall now shew that an infinite determinant converges, provided the
product of the diagonal elements converges absolutely and the sum of th^e non-
diagonal elements converges absolutely.
For let the diagonal elements of an infinite determinant D be denoted
by 1 + aii(i = — X ... + 00 ), and let the non-diagonal elements be denoted
by ttifc ( t > A?, , "* J , so that the determinant is
. . . X *i- C&_i_i ^—10 ^ 11 ■ * *
... ^0—1 X I ^^'^W ^01 • • •
... ^—1 ^0 ■*■ « ^11 • • •
Then since the series
S lott
i^k^-ao
is convergent, the product
00 / 00
p= n (1+ 2
is . 00 \ *•= - 00
is convergent.
a*
I)
Now form the products
m
= n (
1 + 2 a^
m / m
»„= n (1+ 2
O'ik
>
then if, in the expansion of P,„, certain terms are replaced by zero and
certain other terms have their signs changed, we shall obtain Dm ; thus, to
each term in the expansion of Dm there corresponds in the expansion of Pm
a term of equal or greater modulus. Now Dm+p — Dm represents the sum of
those terms in the determinant Dm+p which vanish when the quantities
a{jt {t. A: = + (m + 1) ... ± (m +p)] are replaced by zero ; and to each of these
terms there corresponds a term of equal or greater modulus in Pm+p — Pm*
Hence I i)m^« — D^ I ^ P«i+ti - Pm-
m+p
As the quantities Pm, Pm+u ••• tend to a fixed limit, the quantities Dm,
Dm+i, •.. will therefore tend to a fixed limit. This establishes the proposition.
27, 28] THE THEORY OF ABSOLUTE CONVERGENCE. 37
28. We shall now shew that a determinant, of the convergent form
already considered, remains convergent when the elements of any line are
replaced by any set of quantities whose moduli are all less than som£ fixed
positive number.
Replace, for example, the elements
of the line by the quantities
which satisfy the inequality
I /*r I < M,
where fi is a, positive number; and let the new values of 2)t» and D be
denoted by D^ and D\ Moreover, denote by Pm and P' the products
obtained in suppressing in Pm, and P the factor corresponding to the index
zero ; we see that no term of D^ caai have a greater modulus than the cor-
responding term in the expansion of /JbPm ] and consequently, reasoning as
in the last article, we have
which establishes the result stated.
Example, Shew that the necessary and sufficient condition for the absolute conver-
gence of the infinite determinant
1 oj ...
01 1 oj ...
02 1 O3 •••
is that the series
shall be absolutely convergent. (von Koch.)
Miscellaneous Examples.
1. Find the range of values of ;s for which the series
28iIl2^-4 8in*2!+8sin«2-...+(-l)* + l2*sin*•2+...
is convergent.
2. Shew that the series
1 _ Jl^ _1 1_
is semi-convergent, except for certain exceptional values of z ; but that the series
1+ L+...+ _L_ \ L__ ___L_+_L_+...
t z-^-X '" z-\-p^l z-k-p z-^-p-^-l '" z-h2p-^q-\ z+2p'\-q "*'
in which (p-^q) negative terms always follow p positive terms, is divergent. (Simon.)
38
THE PROCESSES OF ANALYSIS.
[chap. II.
3. Shew that the series
1* 2^ 3* 4^
(l<a<i3)
is oouvergent
4. Shew that the series
is convergent.
5. Shew that the series
(Cesaro.)
a+i3«+a3+/3*+...
(0<aO<l)
(Cesaro.)
7ig*-i
2
((-3"-'}
2kiw
m
converges absolutely for all values of «, except the values
(a=0, 1; k=0, 1, ...m-1 ; m«=l, 2, ... x).
6. If 8^ denote the sum of the first n terms of a convergent series whose sum is «,
shew that
7. In the series whose general term is
u^^qn-Pa: ^ , (0<q<l<x)
where y denotes the number of figures in the expression of n in the ordinary decimal scale
of notation, shew that
i_
Limit tin* =y,
and that the series is convergent, although the quantity ^^^^^ is infinitely great when n is
infinitely great and of the form 1 + lO"- *. (Lerch.)
8. Shew that the series
4
where qn=q'^~n, {0<q<l)
is convergent, although the ratio of the (n + l)th term to the nth is greater than unity
when n is not a triangular number.
9. Shew that the series
(Cesaro.)
00 ^wix
where w is real, and where {w-^-ny is understood to mean c*!**^***), the logarithm being
taken in its arithmetic sense, is convergent for all values of «, when the imaginary part of
X is positive, and is convergent for values of s whose real part is positive, when x is real.
• (— l)n+l
10. Shew that the qth power of the convergent series 2 - ;. is convergent when
ll=sl w
^ — <r, and divergent when ^ — >r.
9 9
(Cajori.)
mSC. EXS.] THE THEORY OF ABSOLUTE CONVERGENCE.
39
11. If the two semi-convergent series
i i^y^ and .<^i|^',
where r and s lie between and 1, be multiplied together, and the product arranged as in
Abel's result, shew that the necessary and sufi&cient condition for the convergence of the
resulting series is r+«>l. (Cajori.)
12. Shew that if the series
be multiplied by itself any number of times, the terms of the product being arranged as
in Abel's result, the resulting series converges. (Cajori.)
13. Shew that the qth power of the series
Oj sin ^+02 sin 2^+ . .. +a« sin n^-f • ..
is convergent whenever ^ <r, r being the maximum niunber satisfying the relation
for all values of n.
14. Shew that if $ is not equal to or a multiple of 27r, and if the quantities
Kq, u^, t£„ ... are all of the same sign and continually .diminish in such a way that the
limit of tfM is zero when n is infinite, then the series Stt^ cos {n6+a) is convergent
Shew also that, if the limit of t^ is not zero, but all the other conditions above are
B , 6 . .
satisfied, the sum of the series is oscillatory if - is commensurable, but that, if - is in-
commensurable, the siun may have any value between certain limits whose difference is
a cosec^^ where a is the limit of u^^ when n is infinite.
(Cambridge Mathematical Tripos, 1896, Part I.)
15. Prove that
i{('-0
.*-i
s>
+ »*-^^+.+
^1
}■
where k is any positive integer, converges absolutely for all finite complex values of z.
16. Let 2 ^M be an absolutely convergent series. Shew that the infinite determinant
A(C) =
(c-4)»-(?o -<?,
-6,
4?-e,
(c-2)»-<»o
2»-<?o'
-61
2»-tf,
4»-(9.
4»-<?<,
2«-tf«
-0.
4«-tfo
0»-A
-6y (c+2)»-Oo
4»-A
4«-(?„
-^4
2»-tfo
(c + 4)»-^o
converges : and shew that the equation
ia equivalent to the equation
A(c)-0
sin' Jwc — A (0) sin* ^ir^o •
(HilL)
CHAPTER III.
The Fundamental Properties of Analytic Functions ;
Taylor's, Laurent's, and Liouville's Theorems.
29. The dependence of one complex number on another.
The problems with which Analysis is mainly occupied relate to the
dependence of one complex number on another. If z and f are two complex
numbers, so connected that the value of one of them is determined by the
value of the other, e.g. if f is the square of z, then the two numbers are
said to depend on each other.
This dependence must not be confused with the most important case of
it, which will be explained later under the title of analytic functionality.
If ^ is a real function of a real variable z^ then the relation between ( and z, which
may be written
f=/(«),
can be visualised by a curve in a plane, namely the locus of a point whose coordinates
referred to rectangular axes in the plane are (z, C), No such simple and convenient
geometrical figure can be found for the purpose of visualising an equation
considered as defining the dependence of one complex number f=f +ii; on another
complex number z=x-\-iy. A representation strictly analogous to the one already given
for real variables would require four-dimensional space, since the niunber of quantities
f > »7i ^1 y? is now four.
One suggestion (made by Lie and Weierstrass) is to use a doubly-manifold systeln of
lines in the quadruply-manifold totality of lines in three-dimensional space.
Another suggestion is to represent £ and 17 separately by means of surfaces
A third suggestion, due to Heffter*, is to write
then draw the surface r^r{x,y) — which may be called the modular-surface of the
function — and on it to express the values of B by surface-markings. It might be
possible to modify this suggestion in various ways by representing B by curves drawn
on the surface r=^r{x, y).
* ZeitschriftfUr Math. u. Phys. xliv. (1899), p. 236.
29, 30] THE FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS. 41
30. Continuity,
Let f(z) be a quantity which, for all values of z lying within given
limits, depends on z.
Let Zi be a point situated within these limits. Then f(z) is said to be
continuous at the point -^i, if, corresponding to any given positive quantity €,
however small, a finite positive quantity 17 can be found, such that the
inequality
i/w-/(*,)i<«
is satisfied so long as | ^ — ^1 { is less than 1;.
U f(z) is continuous at z = Zi, and if its real and imaginary parts be
denoted by u and v, then u and v depend continuously on z.
For if f(z) = w + iv, we have
I (u - 1^) + i (v - Vi) I < 6,
and so {u — v^y -f (v — v^Y < €*,
which gives {u — v^y < e^ and {v — ViY < e^,
and so | u — -Mj | < e and \v — Vi\<€.
The popular idea of continuity, so far aa it relates to a real variable ( depepding on
another real variable z, is somewhat different to that just considered, and may perhaps
best be expressed by the definition " The quantity f is said to depend continuously on z
H as z passes through the series of all values intermediate between any two adjacent
values Zi and z^, ( p&sses through the series of all values intermediate between the
corresponding values d and fg-"
The question thus arises, how far this popular definition is equivalent to the analytical
definition given above.
Cauchy shewed that if a real variable (, depending on a real quantity «, satisfies the
analytical definition, then it also satisfies what we have called the popular definition.
But the converse of this is not true, as was shewn by Darboux. This fact may be illus-
trated by the following example*.
Let B(^) denote the integer next less than x ; and let
f(s)^.[l-E |^-^-^}]+^{j^} si
IT
sm^^
At a;=0, we have/(^)=»0.
Between 4?= - 1 and 47= + 1 (except at a;=0), we have
/(;r)=sin£.
From this it is easily seen that/(^) depends continuously on x near a;=:0, in the sense
of the popular definition, but is not continuous in the sense of the analytical definition.
* Dae to MansioD, Mathens^ ix. (1899).
42 THE PROCESSES OF ANALYSIS. [CHAP. III.
31. Definite integrals.
Let Zo and Z be any two values of z ; and let their representative points
A and B in the ^-plane be connected by an arc (straight or curved) AB;
and let Zi, z^, Zg, ... Zn be a number of points taken on the line AB in any
manner.
Let f(z) be a quantity which, for variations of z along the arc AB,
depends continuously on z.
Let Zq be any point situated in the interval ZqZi of the curve : let Zi be
/ any point situated in the^||ntervap^i2:,: and so on: and consider the sum
S ^f{z^){z, - Z,)'^f{z^){z, - -^0 + ... +/(0(^ - ^n).
We shall shew that if the number n increases indefinitely^ in such a way
that each of the quantities \ z^ — -?r-i I tenis to zero, then this sum wiU tend to
afia>ed limit, independently of the way in which the points
Z\i Z^y ••• Zn, Zq , Zi f ... Zn t
are chosen.
For let € be a given small positive quantity. Since f{z) is continuous,
for each point z=^a of the arc AB we can find a quantity 7}a such that
\f(z) -/(a) I < e.
80 long as \z — a\<rfa'
Let 7} be the least value of rja corresponding to points a on the arc AB.
We shall suppose the subdivision of the arc has been carried so far that
each quantity Ur — '^r-i| is less than tf, and shall first find the effect of
putting in further subdivisions.
Suppose then that the interval ZqZi is subdivided at points z^, z^, "*z^^\
that the interval z^z^ is subdivided at the points ^h> '^i9» ••• Zir^\ and so on :
so that the sum s becomes
«' =f(Zo")(Zoi - Zo) -^f(Zoi)(Zoi - ^oi) + . . .
+/(0(^ii - ^i) +f{Zu){z,^ - ^ii) + . . .
i" • • • »
where z^' is any point in the interval z^z^, z^( is any point in the interval
ZnZoi, and so on.
Then
s-s^ {f(zn ^f(Zo')] (^01 - ^o) + {f(zoO -/(V)! (^« - -^0,) + . . .
+ l/(^i") -fM] (^n - Z,) + {/(^„') -/(^/)j {z,,^Zu) + ...
^r • • . .
31] THE FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS. 48
Therefore
|«'-«| <€ {|^W-^o| + |'2oJ-'^Ol|+ •••}
< € X the length of the broken line connecting the points -e^o» % i -^^w* • • •
where I is the length of the arc AB,
Now by making € indefinitely small, we can make the right-hand side of
this equation as small as we please; and therefore the sum s tends to a
definite limit when the number of subdivisions is indefinitely increased,
provided that at each change in the subdivisions the old points of division
are retained.
The restriction contained in the last phrase has still to be removed.
To do this, suppose that two different methods of division, in eaxih of which
the quantities \Zr — Zr^i\ are less thfin 17, furnish sums s^ and «9. Now
combine the two methods of division, so that every point of division in
either of the original sjchemes becomes a point of division in the new
scheme. Let the sum corresponding to this new method of division be Sj^,
Then since by the above
I «i — «ia I < €/ and I «s — «u I < ely
we have I «i — *a I < 2€Z,
which shews that «i and $2 tend to the same limit. The theorem is thus
established.
The limit thus shewn to exist is called the definite integral of f{z),
taken along the arc AB\ it is denoted by
I
/:
f{z)dz^
AB
in cases where there is no ambiguity as to path, it may be denoted by
f(z) dz.
As an example* of the evaluation of a definite int^;ral directly from the definition,
suppose it is required to find the definite integral of the continuously dependent quantity
(1 -«*)"*, taken along the straight line (part of the real axis) joining the origin (««0)
to a point z=Z^ where Z is real Denote the definite int^;ral by /. Then by definition,
/= Limit 2 ^^'"^^
and the mode of choosing the points z^ and V is arbitrary, within the limits already
explained ; we shall take
«,i=sinrd,
V = sin(r+i)d,
where d= r sin"* Z.
n+l
• Netto, Zeitschrift/Ur Math. xl. (1896).
44 THE PROCESSES OF ANALYSIS. [CHAP. III.
Thus /-Limit I ^^^(r+l)b-^smrb
n ^
= Limit 2 ^ sin 3
= Limit 2 (n+ 1) sin 5
. b
sm-
= sin ~^^ Limit 1
2
=8in-iZ.
The value of the definite integral is therefore sin"^ Z,
32. Limit to the value of a definite integral.
Let M be the greatest value of !/(^)| at points on the arc of inte-
gration AB,
Then |/(V) {zi - z,) -\-f{z^) {z, - ^,) + • • • +/( V) {Z - Zn) \
<|/(V)l|^i--^o| + |/(Olk2--^i|+...+|/(V)|l^-^n|
^ ilf {I ^1 - £-0 I -h i ^, - -?i I -h ... + I ^- -^n '}
where I is the length of the arc of integration AB.
We see therefore, on proceeding to the limit, that
f{z)dz
AB
cannot be greater than the quantity ML
33. Property of the elementary functions.
The reader will be already familiar with the word function^ as used
(in text-books on Algebra, Trigonometry, and the Differential Calculus) to
denote analytical expressions depending on a variable z ; such for example as
z^, c*, \ogz, svxr^z^.
These quantities, formed by combinations of the elementary functions of
analysis, have in common a remarkable property, which will now be investi-
gated.
Take as an example the function e^.
Write e* =f{z).
Then if z' be a point near the point z, we have
z — z z — z z - z
I
( z-/ (z-zY )
32 — 35] THE FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS. 45
and benoe, if the point z tends to coincide with z, the limiting value of the
quotient
z' -z
is (F.
This shews that ^e limiting value of
f{z'y^f{z)
z'^z
%8 in this case independent of the direction of the short path by which the
point / 7noves towards coincidence with z, i.e. it is independent of the
direction in which / lies as viewed from z.
It will be found that this property is shared by all the well-known
elementary functions ; namely, that if f{z) be one of these functions and
h be any small complex quantity, the limiting value of
Jl/(^ + A)-/(^)}
is independent of the mode in which h tends to zero.
34. Occasional failure of the property.
For each of the elementary functions, however, there will be certain
points z at which this property will cease to hold good. Thus it does not
hold for the function at the point z = a, since the limiting value of
z ^ a
If 1^ 1^
h z — a — h z — a]
is not finite when z=^a. Similarly it does not hold for the functions log^
and z^ at the point z = 0.
These exceptional points are called singular points or singularities of the
function f(z) under consideration ; at other points the function is said to be
regular,
36. The analytical function.
The property noted in § 33 will be taken as the basis of our definition of
an analytic function, which may be stated as follows.
Let an area in the -e-plane be given ; and let w be a quantity which has
a definite finite value corresponding to every point z in that area. Let
z, z-^-izhe values of the variable z at two neighbouring points, and UyU-\rhu
the corresponding values of u. Then if at every point z within the area
^ tends to a finite limiting value when hz tends to zero, independently of
46 THE PROCESSES OF ANALYSIS. [CHAP. Uh
the way in which Bz tends to zero, u is said to be an analytic function of z,
regular within the area.
We shall generally use the word " function " alone to denote an analytic
function, as the functions studied in this work will be almost exclusively
analytic functions.
In the foregoing definition, the function u has been defined only within
a certain area in the ^-plane. As will be seen subsequently, however, the
function u will generally exist for other values of z not excluded in this area :
and (as in the case of the elementary functions already discussed) may have
Angularities, for which the fundamental property no longer holds, at certain
points outside the limits of the area.
The definition of functionality must now be translated into analytical
language.
If /(^) be a function of z, regular in the neighbourhood of a particular
value Zy then, by the definition, the quantity
z — ^
tends to a definite limit, depending only on z, when / tends to z. Let this
limit be denoted by the symbol /' (z).
Then (by the definition of a limit) for every positive quantity 6, however
small, it is possible to find a quantity 17, such that
is less than €, so long as \z — z\ is less than 17.
If therefore we write
/(/) =f{z) + (/ - Z)f' {Z) + 6 (/ - Z\
we see that | e' | is less than e, so long as | / — ^ | is less than 17 ; that is, the
function /(-gr) must be such that the quantity e', defined by the equation
/(/) =/(^) + (/ - z)f (^) + « (/ - z\
tends to the limit zero as z tends to z.
The necessity for a strict definition of the term "function" may be seen from the
following consideration.
Let y denote the temperature at a certain place at time U As t varies, y will vary,
and y may loosely be called a "function" of t But y cannot be expressed in terms of t
by a Maclaurin's infinite series
'-w.-.-^'(l),./^(S),
I • • • >
** \w*V<-0
for if it could, the knowledge of the temperature for a single day would enable us to
determine the quantities
36j THE FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS. 47
and then from the Maclaurin's expansion it would be possible to predict the temperature
for the future !
Maclaurin's series is in fact, as will appear subsequently, applicable only to analytic
functions, in the sense in which analytic functions have been defined above.
36. Cauchys theorem on the integral of a function round a contour.
A simple closed curve in the plane of the variable z is often called
a contour : if -4, jB, C, D be points taken in order along the arc of the
contour, and i{f(z) be a quantity depending on z and continuous at all points
on the arc, then the integral
/
f(z)dz,
ABCDA
taken round the contour, starting from the point A and returning to A again,
is called the integral of the quantity f(z) taken round the contour. Clearly
the value of the integral taken round the contour is unaltered if some point
in the contour other than A is taken as the starting-point.
We shall now prove a result due to Cauchy, which may be stated as
foUowa If f(z)i8 an analytic function, regular at all points in the interior
of a contour, then
I
Az)dz=0,
where the integration is taken round the contour.
For let A, B, (7, D be points in order on the contour. Join A to C hy an
arc AECy which will divide the region contained within the contour into two
distinct portions. Then the integral taken round the contour ABCDA is
equal to the sum of the integrals taken round the two contours ABCEA and
ABCDA ; for
f f{z)dz+{ f{z)dz
J ABCEA J ABCDA
= f f{^)dz+j f(z)dz+f f(z)dz-^j f(z)dz
J ABC J CEA J ABC J CD A
= f f{z) dz.
J ABCDA
since the integrals along CEA'^ and AEC neutralise each other.
Now join any point E on the arc AEC to D by an arc EFD, and join
^ to jS by an arc EOB; then in the same way we see that the integral
round ABCEA is equal to the sum of the integrals round ABOEA and
EOBCE, and the integral round AECDA is equal to the sum of the
integrals round AEFDA and DFECD.
Thus the original contour-integral is equal to the sum of the integrals
48 THE PROCESSES OF ANALYSIS. [CHAP. III.
round the four contours ABGEA, EGBCE, AEFDA, DFECD, into which it
has been divided by drawing the cross-lines.
Proceeding in this way by drawing more cros^-lines, we see that the
original contour-integral can be decomposed into the sum of any number of
integrals round smaller contours, which constitute a network filling up the
original contour.
Now suppose that each of these small contours has linear dimensions of
the same order of magnitude as a small quantity l. Let z^^ be a point within
one of them. Then on this small contour we have
f{z) ^f(Zo) + (Z- Zo)f (Z,) + (^ - Zo) €,
where e is infinitely small when I is infinitely small.
Thus ^f{z)dz^^f{z,) dz-^-jiz - Zo)f{z,) dz+j{z--Zo)€dz,
where all the integrals are taken round the small contour.
Now j / (zo) dz =f(zo) j dz
=/(zq) X the increase in value of z after once*
describing the small contour
= 0.
Similarly j/(zo) (z - z^) dz = lf(Zo)jd {(z - Zof] = 0,
when the integral is taken round the small contour.
Thus, if 7} be the greatest value of | e | for points on the small contour,
we have
jf(z)dz^^rfl\z-Zo\\dz\,
where the integrals are taken round the small contour.
Now the right-hand side of this equation is clearly of the order rjl^ of small
quantities. The value of jf(z)dz, taken round the small contour, is there-
fore a small quantity of order tflK
Now the number of such small contours ip a given area is of the order
1 . '
T^. If 77' be the maximum value of 7} for all the small contours in the
area, we see therefore that the total sum of the integrals for all the small
contours in the area is at most of the order r)'l^ Xj^ or 1;'; and 17' can be
made indefinitely small by decreasing I.
It follows, therefore, that the sum of the integrals round all the small
36] THE FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS. 49
contours is zero; that is, the integral round the original contour is zero,
which establishes Cauchy's result.
Corollary 1. If there are two paths ZqAZ and ZoBZ from Zq to Z, and if
f(z) is a regular function of z at all points in the area enclosed by these two
rz
paths, then / f(z)dz has the same value whether the path of integration is
z^AZ or ZqBZ. This follows from the fact that ZqAZEzq is a simple contour,
and so the integral taken round it (which is the diflference of the integrals
along z^AZ and z^BZ) is zero. Thus, i{f{z) be an analytic function of z, the
value of I f(z) dz is to a certain extent independent of the choice of the
arc AB, and depends only on the terminal points A and B. It must be
borne in mind that this is only the case when f{z) is an analytical function in
the sense of § 35.
Corollary 2. Suppose that two simple closed curves C^ and Ci are given,
such that Cq completely encloses Cy, as e.g. would be the case if Ci and Ci
were coucentric circles or confocal ellipses.
Suppose moreover that f{z) is an analytic function, which is regular at
all points in the ring-shaped space contained between C^ and (7i. Then by
drawing a network of intersecting lines in this ring-shaped space, we can
shew exactly as in the theorem just proved that the integral
ff(z)dz
is zero, where the integration is taken romfid the whole boundary of the ring-
shaped space; this boundary consisting of two curves Co and Ci, the one
described in a positive (counter-clockwise) direction and the other described in
a negative {clockwise) direction.
Corollary S. And in general if any connected region be given in the
^- plane, bounded by any number of curves C?o, Ci, C„ ..., and if f(z) be
any function of z which is regular everywhere in this region, then
j/(z) dz
is zero, where the integral is taken round the whole boundary of the region; this
boundary consisting of the curves Cq, Ci, ... , each described in such a sense that
the region is kept either always on the right or always on the left of a person
walking in the sense in question round the boundary.
An extension of Cauchy's theorem I f{z)dz=0, to curves lying on a cone whose vertex
is at the origin, has been made by Raout {Nouv. Annales de Math. (3) xvi. (1897),
w A. 4
50 THE PROCESSES OF ANALYSIS. [CHAP. IIL
pp. 365-7). Osgood {BvU. Amer, Math, Soc,, 1896) has shewn that the property jf{z) dz—0
may be taken as the defining-property of an analytic function, the other properties being
deducible from it.
Example. A ring-shaped region is bounded by the two circles U| = l and U|=2 in the
/dz
— , where the integral is taken round the boundary
of this r^on, is zero.
For the boundary consists of the circumference |«| = 1, described in the clockwise
direction, together with the circumference !«| = 2, described in the counter-clockwise
direction. Thus if for points on the first circumference we write «=«**, and for points on
the second circumference we write z=2e^, then B and (j) are real, and the integral becomes
-«» t.^de . f^ i.2e^d4>
2e^
Jo ^' ^Ji
or -2frt + 2fri, i.e. zero.
37. The vaiiie of a function at a pointy expressed as an integral taken
round a contour enclosing the point.
Let (7 be a contour within which f(z) is a regular function of z.
Then if a be any point within the contour, the expression
z— a
represents a function of z, which is regular at all points within the contour C
except the point z^a, where it has a singularity.
Now with the point z^sa as centre, describe a circle 7 of very small
radius. Then in the ring-shaped space between 7 and C, the function
/(f)
z — a
is regular, and so by Corollary 2 of the preceding article we have
f fiz)dz r f(s)dz ^^^
J c Z'-a J y z — a
where I and I denote integrals taken in the positive or counter-clockwise
sense round the curves C and 7 respectively.
But (§35) f/(fM^,f/(«)-K^-")/'(«) + '(^-«)rf,
^ Jy z-a Jy z—a '
37, 38] THE FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS. 51
where € is a quantity which tends to zero when the radius of the circle 7 is
indefinitely diminished. Thus
Jcz-a -^^^lyZ-a Jy Jy
Now if at points on the circumference 7 we write
z — a- re^,
where r is the radius of the circle 7, we have
f dz P^ir^^de . r^' ,^ ^ .
and f dz=^jire^d0 = O;
I f I
also I edzl^rj. iirr,
where 17 is the greatest value of |€| for points ^ on 7; and therefore in the
limit when r is made indefinitely small we have
/.
edz = 0.
y
"- L4^-2"/<»).
C
or
•^ ^ ^ 27n J c z — VL
This remarkable result expresses the value of a function /(^) at any point
a within a contour (7, in terms of an integral which depends only on the value
of f(z) at points on the contour itself.
Corollary, If f(z) is a regular function of ^ in a ring-shaped region
bounded by two curves C and C, and a is a point in the region, then
-^ ^ "^ zmjoz — a imjcz — a
where C is the outer of the curves and the integrals are taken in the positive
or counter-clockwise sense.
38. The Higher Derivates.
The quantity /'(-gr), which represents the limiting value of
f{z + h)-f{z)
h
when h tends to zero, is called the derivaie o{ /(z). We shall now shew that
/' (z) is itself an analytic funcHon of z, and consequently Usdf possesses
a derivate.
4—2
52 THE PROCESSES OF ANALYSIS. [CHAP. HI.
For if (7 be a contour surrounding the point z, and situated entirely
within the region in which f{z) is regular, we have
*=o 27riA \j c Z'-a — h j c z — a \
= J^r/M^+LimitA[ /(^) ^^
Now f /(^)^^
J c(z — ay{z — a-'h)
is a finite quantity, since the integrand
(z-ay{z-a-h)
is finite at all points of the contour (7, and the path of integration is of finite
length. Hence
Limit ^-; <^ rT = 0»
and consequently /' (a) = ^ j ^^^ >
a formula which expresses the value of the derivate of a function at a point
as an integral taken round a contour enclosing the point.
From this formula we have, if h be any small quantity,
f (a + h) -f (g) 1 [ mdz \ 1 1_)
h 2m} c h \{z-a-hy {z-af)
2mJc {z-a-hy(z-ay
zmj c {z — o)
where ^ is a quantity which is easily seen to remain finite as h tends to zero.
Therefore as h tends to zero, the expression
/'(a+h)-f'ia)
h
tends to a limiting value, namely
2 /■ /{z) dz
2'iTiJc{z— ay '
38] THE FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS. 63
The quantity /' (a) is therefore an analytical function of a ; its derivate,
which is represented by the expression just given, is denoted by /" (a), and is
called the second derivate of /(a).
Similarly it can be sheAvn that f" {a) is an analytical function of a,
possessing a derivate equal to
3 r f{z) dz ^
iJciz-a/'
this is denoted by /'" (a), and is called the third derivate of /(a). And in
general an nth derivate of/ (a) exists, expressible by the integral
n\ f f (z) dz
and having a derivate of the form
(n+1)!
l)!r f{z)dz ,
27rt ic{z-ay^'
this can be proved by induction in the following way.
Then
/(»)(a + A)-/(») (a) n\ [ J(z)dz ( 1 1 ]
h i-rnio h l(«-a-A)»+' (^-a)*+'j
^ n!_f f{z)dz U h \-»->_,)
2iriJc(^-a)"*'AlV e-a) )
(n + l)! f f{z)dz
+ terms which vanish when h tends to zero.
which establishes the required result.
A function which possesses a first derivate at all points of a region in the
^-plane therefore possesses derivates of all orders.
Exam'ple 1. Verify the theorem
by use of Taylor's Theorem.
By Taylor's Theorem we have
54 THE PROCESSES OF ANALYSIS. [CHAP. III.
f dz
But when i: is an integer other than unity, I ^ _ ^ is zero, since
resumes
its original value after describing the contour. So the only surviving part of the right-
hand side is -zr- ./H (a) / , or/(*) (a).
Example 2. Verify the same theorem by means of integration by parts.
We have
nl[ f{z)dz _ ( {n-\)\ f{z) \ (n-iy. f r{z)dz
f(z)
and the first term is zero, since 7-- ^. resumes its original value when z makes the circuit
(z—ap
of the contour C. Proceeding in this way, we have
n! f f(z)dz ^ 1 f f!!Hl)dz
39. Taylor's Theorem.
CoDsider now a function f{z\ which is regular in the neighbourhood
of a point z^a. Let C be the circle of largest radius which can be drawn
with a as centre in the ^-plane, so as not to include any singular point of
the function f{z)\ so that f{z) is a regular function at all points of G. Let
5sa + A be any point within the circle G. Then by §37, we have
^/ IN If f{e)dz
•' ^ ^ zm Jc z — CL — h
But at points z on the circle G. the modulus of ^ , will not exceed
^ z—a—h
some finite quantity M,
Therefore •
1 r /{zydz.h""^^ M.^irR /[A_|\«+i
27riJa(^-a)«+^(^-a-A) ^ 2ir \RJ '
where R is the radius of the circle G, so that 27ri2 is the length of the path
of integration in the last integral, and R^\z — a\ for points z on the cir-
cumference of G,
The right-hand side of the last inequality tends to zero as n increases
indefinitely. We have therefore
39] THE FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS. 56
which we can write
«
This- result is known as Taylor's Theorem] the proof we have given is due
to Cauchy, and shews exactly for what range of values of z the theorem
holds true, namely for all points z which are nearer to a than the nearest
singularity of/ {z). It follows that the radios of convergence of a power-aeriee
%8 always such as jtist to exclude from, the circle of convergence the nearest
singularity of the function represented by the series.
At this stage we may introduce some terms which will be frequently
used.
If/(a) = 0, the function /(-^) is said to have a zero at the point -^ = a.
If at such a point/' (a) is diflferent from zero, the zero of /(a) is said to be
simple; if, on the other hand, the quantities /'(a), /''(a), .../<**~^* (a) are all
zero, so that the Taylor's expansion of f(z) at ^ = a begins with a term
in {z — ay*, then the function f(z) is said to have a zero of the nth order at
the point z=^a.
Example 1. Find a functiou /(«), which is regular within the circle C of centre at the
origin and radius unity, and has the value
a-oos^ sin^
a«-2aco6^+l a*-2acos^+l
(where a>\ and B is the vectorial angle) at points on the circumference of C,
We have
f{z) dz
c ^•^^
« **• . tad . —3 — r :r-— , puttmg «■■«•*
^j, since the only non-zero term is that from k^n.
a^
Therefore by Maclaurin's Theorem*,
or /(«)«- • for all points within the circle.
This example raises the interesting question, What isf(z) for points outside the circle?
Is it still — ? This will be discussed m §§ 41, 42.
«0
Example 2. Prove that the arithmetic mean of all values of «"* I a^«f, for points «
on the circumference of the circle |£|«1, is a«, if Sa^^*" is regular at all points within the
circle.
♦ The result /W=/(0) + */'(0)+^/"(0)+....
2
which is obtained by putting asO in Taylor's Theorem, is osoally called Maclaurin*i Theorem,
56 THE PROCESSES OF ANALYSIS. [CHAP. III.
• f{v) (0)
Let 2 avz^=if{z)y so that a^^ — \- . Then the required mean is
— • I —^^Tr- J where G is the circle,
or
27rt, ,
/(H) (0)
or J \ I
ni
or a,j.
f »
Example 3. Prove that if A is a given constant, and (1 - 2zh+h^)~^ is expanded in the
form
l+hP^(z)+h^Pi{z)-{-¥P^(z) + (A),
where F^ {z) is easily seen to be a polynomial of degree n in Zy then this series converges so
long as ; is in the interior of an ellipse whose foci are the points s=l and z^-l, and
whose semi-major axis is 5 (A+ j) .
Let the series be first regarded as a function of h. It is a power-series in A, and
therefore converges so long as the point h lies within a circle on the A-plane. The centre
of this circle is the point A=0, and its circumference will be such as to pass through that
singularity of (1 -2aA+ A*)"* which is nearest to A=0.
But l-2zh+h^=(h'Z+^^^) (A-«- V^^-^l),
so the singularities of (1-2M4-A*)"* are the points A=«-(«*-l)* and A««+(«*-l)*,
at which it is infinite.
Thus the series (A) converges so long as |A| is less than either
U-(2«-l)*| or !«+(«»- 1)*|.
Now draw an ellipse in the z-plane passing through the point z and having its foci at
the points 1 and - 1. Let a be its semi-major axis, and 6 the eccentric angle of z on it.
Then z=acoB6+i{a*-'l)^Bin6y
which gives z±{s^- l)*={a±(a*- 1)*} (cos d+t sin 6\ •
so |«±(22-l)*|-a±(a8-l)*.
Thus the series (A) converges so long as A is lees than the least of the quantities
a'^{a^- 1)* and a-{a^— 1)*, i.e. so long as A is less than a—ia^- 1)*. But
A=a— (a*-l)* when <*=3(^+i)-
Therefore the series (A) converges so long as ^ is within an ellipse whose foci are I and
- 1, and whose semi-major axis is - I^+t ) •
40. Forma of the remainder in Taylors Series,
The form found in the last article for the remainder after n terms in
Taylor's series is
f(z) JC^dz
40, 41] THE FUNDAMENTAL PBOPERTIES OF ANALYTIC FUNCmONS. 57
It is not difficult to derive from this expression the forms of the remainder
usually given in treatises on the Diflferential and Integral Calculus. For
on mtegrating by parts the quantity n I ^>— — _ .x^-h , we have
Jo (z-a- 1)"^^ {z - a)«+^ ^Kn-i-^) ]^ ^^ ^a^ty^^
k^ h^
1
"• / - " _ W4.« •"•••>
{z - a)»*+i {z - a)'»+«
by successive repetition of this process,
A« '
"" (2r — aY{z -a-hy
which is a new form for the remainder.
Now suppose that all the quantities concerned are real. Then along the
line of integration, {h — ^)**~^ has a fixed sign, so
^»=(^?i)i/„*<'^-'>"-''^'
where H lies between the greatest and least values of/<**>(a + ^) between
< = and t = k We can therefore write H =/<*»> (a + Oh), where < d < 1,
and then
or iJ = ^/(»)(a + ^A),
which is Lagrange's form for the remainder,
Darboux gave in 1876 {Journal de Math. (3) ii. p. 291) a form for the remainder in
Taylor's Series, which is applicable to complex variables and resembles the above form
given by Lagrange for the case of real variable&
41. The Process of Continuation,
Near every point P{z^ at which a function f{z) is regular, we have
seen that there is an expansion for the function as a series of ascending
positive integral powers of {z — z^, the coefficients in which are the suc-
cessive derivates of the function at z^.
Now let A be the singularity o{ f(z) which is nearest to P, Then the
circle within which thus expansion is valid has P for centre and PA for
radius.
58 THE PROCESSES OF ANALYSIS. [CHAP. in.
Suppose that we are given the values of the function at all points of the
circumference of this circle, or more strictly speaking, of a circle slightly
smaller than this and concentric with it : then the preceding theorems enable
us to find its value at all points within the circle. The question arises, How
can the values of the function at points ouUide the circle be found ?
In other words, given a power-series which converges and represents a
fwnction only at points within a circle, to derive from it the values of the
function at points outside the circle.
For this purpose choose any point Pi within the circle, not on the line
PA, We know the value of the function and all its derivates at P,, from
the series, and so we can form the Taylor series with Pi as origin, which
will represent the function for all points within some circle of centre P,.
Now this circle will extend as far as the singularity which is nearest to Pi,
which may or not be A ; but in either case, this new circle will generally* lie
partly outside the old circle of convergence, and for points in the region
which is included in the new circle hut not in the old circle, the new series tvill
furnish the values of the function, although the old series failed to do so.
Similarly we can take any other point P^, in the region for which the
values of the function are now known, and form the Taylor series with P,
as origin, which will in general furnish the values of the function for other
points at which its values were not previously known ; and so on.
This method is called continuation'^. By means of it, starting from a
representation of a function by any one power-series we can find any number
of other power-series, which between them furnish the value of the function
at all points where it exists ; and the aggregate of all the power-series thus
obtained constitutes the analytical expression of the function.
Example, The aeries
represents the function
1 £ £« «3
' ^ ' a-z
only for points z within the circle \z\=a.
But any number of other power-series exist, of the type
1 , r~6 (z-by (z-bf
a-6"^(a-6)2'*"(a-6)s'*"(a-6)*"*"'*'*
which represent the function for points outside this circle.
* The word "gfenerallj" must be taken as referring to the cases which are likely to come
under the student's notice before he reads the more adyanoed parts of the subject,
t In Qerman, Fortsetzung.
42] THB FUNDAMENTAL PROPERTIES OF ANALYTIC FCTNCTIONS. 59
On functions to which the continuation-process cannot be applied.
It is not always possible to carry out the process of continuation. Take as an example
the function f{z) defined by the power-series
/(«)=1 +««+«♦ -f^^+^W-H.. .+««" + ...,
which clearly converges in the interior of a circle whose radius is unity and whose centre
is at the origin.
Now as t approaches the value +1 by real values, the value of /(«) obviously tends
towards -foo ; the point +1 is therefore a singularity of f{z).
But /W=^*+/W,
80 if « is such that -?*=1, and therefore /(«*) is infinite, then f(z) is also infinite, and so
« is a singularity off{z) : the point «=■ - 1 is therefore a singularity of /(«).
Similarly since
we see that if « is such that z*— 1, then zis & singularity oif{z) ; and in general, any root
of any of the liquations
«8=1, 5*-l, «8-l, «i«-l, ...,
is a singularity of f(z). But these points all lie on the circle |«| = 1 ; and in any arc
of this circle, however small, there are an infinite number of them. The attempt to
carry out the process of continuation will therefore be frustrated by the existence of this
unbroken front of singularities, beyond which it is impossible to pass.
In such a case the function f(z) does not exist at all for points z situated outside the
circle \z\»l ; the circle is said to be a limiting circle for the function.
42. The identity of a function.
The two series
1 + ^-1- ^« + ^+...
and - 1 + (2: - 2) - (-2 - 2)« + (-? - 2)» - (^ - 2)* + . . .
are not simultaneously convergent for any value of z, and are distinct
expansions. Nevertheless, we generally say that they represent the same
functiony on the strength of the fact that they can both be represented by the
same rational expression .
This raises the question of the identity of a function. Under what
circumstances shall we say that two different expansions represent the sa/me
function ?
We shall define a function, by means of the last article, as consisting of
one power-series together with all the other power-series which can be
derived from it by the process of continuation. Two diflferent analytical
expressions will therefore be regarded as defining the same function if they
represent power-series which can be derived from each other by continuation.
It is important to observe that a single analyticcU eicpression can represent
different functions in different parts of the plane. This can be seen from the
following example.
60 THE PROCESSES OF ANALYSIS. [CHAP. IH.
Consider the series
The sum of the first n terms of this series is
1 / 1\ 1
The series therefore converges for all finite values of z. But since when
n is infinitely great, z^ is infinitely small or infinitely great according as | -? |
is less or greater than unity, we see that the sum to infinity of the series is
z "when |-2r|<l, and - when 12:|>1. This series therefore represents one
z
fwnction at points in tlie interior of the circle | ^r | = 1, and an entirely different
ftmction at points outside the same circle.
Example. Shew that the series
« + g2 + ^+ ,^2+ ...
, 1 / 2z 2 1 / 2zy .241 / 2g V \
2\l-^ 3"3- Vl-«V 3;6-6' Vl-zV -J
represent the same function in the common part of their domain of convergence.
43. Laurent's Theorem,
A very important extension of Taylor s Theorem was published in 1848
by Laurent ; it relates to the expansion of functions under circumstances in
which Taylor s Theorem cannot be applied.
Let C and C" be two concentric circles of centre a, of which C is the inner;
and let f{z) be a function which is regular at all points in the ring-shaped
space between C and G\ Let a + A be any point in this ring-shaped space.
Then we have (§ 37, Corollary)
y(. + A) l.f -Af) ,, l_.r _/(?L d,,
•^ ^ ^mjcz — a — h 2'mJcZ^a — h
where the integrals are supposed taken in the positive or counter-clockwise
direction round the circles.
This can be written
+
. 43] THE FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS. 61
We find, as in the proof of Taylor's Theorem, that
[ f{z)dz,h^^' f f(z)dz{z-a)^
tend to zero as n increases indefinitely ; and thus we have
b b
where a^^^f^I^, and K ^ ^. f J^ - aT'^ (z) dz.
This result is Laurent* 8 Theorem; changing the notation, it can be
expressed in the following form : If z be any point in the ring-shaped space
within which f{z) is regular, and which is bounded by the two concentric
circles C and C" of centre a, then f (z) can be expanded at the point z in the
form
h h
An important case of Laurent's Theorem arises when there is only one
singularity within the inner circle C", namely at the centre a. In this case
the circle C can be taken to be infinitely small, and so Laurent's expansion
is valid for all points in the interior of the circle C, except the centre a.
Example 1. Prove that
1 f^^
where •^n(^)=5- I coB{n6^xsmB)d3.
For Laurent's Theorem gives
z z
where 0^=5^. f Z^*"^^ :^, and bn=J-. [ J^"'^ ^'^dz,
and where C and C" are any circles with the origin as centre. Taking C to be the circle of
radius unity, and writing z^e , we have
2trtyo
= — I cos (nd - ^ sin d) cW,
62 THE PROCESSES OF ANALYSIS. [CHAP. HI.
since the parts of / sin (n^-a;sin 6)dB which arise from B and ^ir-6 will destroy each
other. Thus
Now ft»= (-!)*«»> since the function expanded is unentered if -- be written for z,
z
Thus
5«=(-i)"y,(x),
which completes the proof.
Example 2. Shew that, in the annulus defined by
\a\<iZi<\bU
the expression -j , . yr — x[ can be expanded in the form
where 5.- S^ i^.n-:mi^. '[l) •
For by Laurent's Theorem if C denote the circle |«|«»r, where |a|<r<|6|, then the
coefficient of z* in the required expansion is
Putting z=re^f this becomes
or
2.r j, * '^ '^l 2*.*! -p-? 201 H~-
The only terms which give integrab different from zero are those arising from k^l+n.
So the coefficient of z^ is
1 p*^^ 1.3. ..(2^-1) 1.3... (2 ^ +2n-l) <^
Sirjo I 2'.^! 2'+-.(";+n)! &'+»'
Similarly it can be shewn that the coefficient of -- is S^a^,
Example 3. Shew that
e'**'*"«-ao+«i«+«8«*+— +- + J+...>
where *'»*^2ir / ^"■*"*'^*^*<^{(^-*^)8"^^-w^}^>
and ^»=o^- I ^'''^^'''*^ooa{{v-' u) sine -n6}de.
^ir J Q
44] THE FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS. 63
44. The nature of the singularities of a one-valued function.
Consider now a function f{z) which is regular at all points of a certain
region in the -e-plane, except a point z = a;so that the point a is a singularity
of the function /{z)*
Surround the point a by a small circle 7, with a as centre. Then in the
ring-shaped space between 7 and some larger concentric circle (7, the function
/ (z) can by Laurent's Theorem be expanded in the form
^0 + -4 1 (^r — a) + ^a (-2^ - a)* + -4 , (^ — a)* + . . .
z --a {z — ay (z — ay
The terms in the last line are called the Principal Part of the expansion of
the function at the singularity a ; if they were non-existent, the function
would clearly be regular at the point ; so they may be regarded as consti-
tuting the analytical expression of the singularity.
Now these terms of the Principal Part may be unlimited in number,
i.e. the series
Si B^ Bi
z — a {Z'-ay {z — ay
may be an infinite series ; in this case the point a is said to be an essential
singularity^ of the function /(z). Or on the other hand, they may be
limited in number, i.e. the series just written down may be a terminating
series ; so that the expansion can be written in the form
Bn Bn-i _^ ^_Bi,^A, + Ai{z''a)'\'A2{z^ay+....
{z-aY {z-af^^ '" z-a
In this case the function is said to have a pole of order n at thelpoint a.
When n is unity, so that the expansion is of the form
— ^H-ilo-hili(-?-a)-hil,(-^-a)"+...,
z — a
the singularity is said to be a simple pole.
Example 1. Find the singularities of the function
c
z
Near 2=0, the function can be expanded in the form
e CM afi
^ a a^ a*
* The name essential singularity is also applied to any singularity of a one-valued ftmotion
which is not a pole, i.e. to singularities for which no Laurent expansion at all can be found.
64 THE PROCESSES OF ANALYSIS. [CHAP. III.
c
e~
or
— ^ e « ( - + c, ) +positive powers of z.
There is therefore a simple pole at 2=0. Similarly there is a simple pole at each
of the points ^trnia {n= +1, +2, +3, ...).
Near 2=0, the function can be expanded in the form
c
gt-a
B-a
14.-1-+ - +
or ^
e
(>^"-?--)-' '
which gives an expansion involving all positive and negative powers of {z - a). So there is
an essential singularity at 2= a.
There is also an essential singularity at 2=00 , as will be seen after the explanations of
the next article.
Example 2. Shew that the function defined by the series
, >^-'{(i+y-i '
has a simple pole at each of the points
/ 1\ ^"^
z^il + -\e * (it«0, 1,2, ...7i-l; ?i=l, 2, ...00).
(Cambridge Mathematical Tripos, Part II., 1899.)
45. The point at infinity.
The behaviour of a function f(z) for infinite values of the variable £
can be brought into consideration in the same way as its behaviour for finite
values of z.
For write ^ = -> , so that the infinite values of z are represented by the
point / = in the /-plane. Let f(z)^<f> (z). Then the function ^ (/) may
have a zero of order m at the point 2:' = ; in this case the Taylor expansion
of ^ (/) will be of the form
and so the expansion oi f{z) valid near z^(x> will be of the fonn
T" • • • •
In this case, / (z) is said to have a zero of order m at z==oo ,
45] THE FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS. 65
Again, the function ^ (/) may have a pole of order m at the point / = ;
in this case,
and so for large values of z,f{z) can be expanded in the form
In this case, ^ = oo is said to be a pole of order m for the function /{jb).
Similarly f(z) is said to have an essential singularity at -^ = oo , if ^ {£)
has an essential singularity at the point z = 0. Thus the fiinction eF has an
1
essential singularity at 5 » oo , since the function ^ or
has an essential singularity at / = 0.
Example, Discuss the function represented by the series
The function represented by this series has singularities at z^--^ and f~ — -y
(n«l, 2, 3, ...), since at eacb of these points the denominator of one of the terms in the
series is zero. These singularities are on the imaginary axis, and are infinitely numerous
near the origin ^=0 : so no Taylor or Laurent expansion can be formed for the function
valid ia the region immediately surrounding the origin.
For values of z other than these singularities, the series converges absolutely, since the
ratio of the (n+l)th term to the nth is ultimately . > ^ , which is veiy small when n
is larga The function is an even function of z (i.e. is unchanged if the sign of ^ be
<:hanged}, is zero for all infinite values of 2, and m regular at all points outside a circle
C of radius unity and centre at the origin. So for points outside this circle it can be
expanded in the form
-where, by Laurent's Theorem,
** 2ni J a n=ow^ «"■** + «*
0-a«^tt-l ^»-3^-2«/ a-2n ^-4n ^-e» \
^^"^ n\(a'^+z^)" ^1 V""i5~ "*""?"" 15-+-J'
1 (-!)*-! a"**
and the coefficient of - on the right-hand side of this equation is ^ — -. .
Z 7Cr •
W. A. 5
66 THE PROCESSES OF ANALYSIS. [CHAP. lU.
Therefore, since only terms in - can fiimish non-zero integrals, we have
*ifc=o— • ^ I — i
^ 2mn^oJc n\ z
j_
=(-l)*-ie«".
Therefore for large values of z (and indeed for all points z outside the circle of radius
unity) the function can be expanded in the form
JL J. J_
The function has a zero of the second order at z== ao , since the expansion begins with
a term in -5 .
z^
46. Many-valued functions.
In all our previous work we have supposed the function f(z) to have one
definite value corresponding to each value of z.
But functions exist which have more than one value corresponding to
each value of z. Thus the function ^ has two values (viz. + VJ and — V^)
corresponding to each value of z, and the function tan~* z has an infinite
number of values, expressed by the formula tan~^ z ± hrr, where k is any
integer.
We may however for many purposes consider + Vz and — VJ as if they
were two distinct functions, and apply to either of them separately the
theorems which have been investigated in this chapter. When we in this
way select some one determination of a many- valued function for considera-
tion, it is called a branch of the mimy-valued function. Thus the values
log z, log z + 27rt, log z + 4iTn, . . . , would be said to belong to different branches
of the function log z.
There will be certain points for which the values of the function given by
diflferent branches coincide: these points are called branch-points of the
function, and must be included among its singularities. Thus the function
jgr* has a branch-point at jgr = 0, since either branch there gives the same value,
zero, for the function.
It must not however be supposed that the branches of a many-valued
function really are distinct functions. The following example shews how
the different branches of a many-valued function change into each other.
Let f{z) = A
46] THE FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS. 67
Write 2:=r(co8^ + i8ind), where 0<^<27r. Then the two values of
f{z) are
e . . e\ , ,-/ . . . 0'
+ Vr [cos ^ + i sin 5] and - Vr [cos ^ + i sin 5 j .
Let us take the former of these values, and consider its changes as the
point z describes a circle round the origin {z = 0). As the point travels, r is
unchanged, but constantly increases, and when the point reaches again the
starting-point after completing the circuit, has increased by 27r. The
function has therefore become
r { + 27r . . ^ + 27r\
-h Vr (cos — 2 — + ^ sin — — j ,
^r — Vr (cos^ + isin ^j.
In other words, the branch of the function with which we started has passed
over into the other branch.
In following the succession of values of f{z) along a given path, the final
value is deduced without ambiguity from the initial value; and all con-
ceivable paths lead to one of two final values, viz. 'Jz and — Vz, But it
appears from the above that it is not possible to keep these branches per-
manently apart as distinct functions, because paths lead from one value to
the other.
The idea of the different branches of a function helps us to understand many of the
''paradoxes'' of mathematics, such as the following.
Ck)nsider the function
du
for which t- =^ (1 +log «).
When z is negative and real, -^ is not real. Now if ^ is a negative quantity of the
form \^ (where p and q are positive or negative integers), u is real.
If therefore we draw the real curve
we have for negative values of « a series of conjugate points, arranged at infinitely small
intervals of z : and thus we may think of proceeding to form the tangent as the limit of
du
the chord, just as if the curve were continuous ; and thus -r- , when derived from the
inclination of the tangent to the axis of a?, would appear to be real The question thus
dtt
arises, Why does the ordiniiry process of differentiation give a non-real value for -r- ? The
5—2
68 THE PROCESSES ^OF ANALYSIS. [CHAP. IH.
explanation is, that these conjugate points do not all arise from the same branch of the
function u=^. We have in fact
and log z has an arbitrary additive p£urt 2lrirt, where k is any integer. To each value of k
corresponds one branch of the function u. Now in order to get a real value of u when z is
negative, we have to choose a suitable value for k : and this value ofk varie$ atvfego from
one conjugate point to an adjacent one. So the conjugate points do not represent values of
u arising from the same branch of the function ubs^*, and consequently we cannot expect
diL
the value of -r- to be given by the tangent of the inclination to the axis of x of the
tangent-line to the series of conjugate points.
Example 1. If log z be defined by the equation
log i?a Limit n{f^—\\
»=•
shew that log 2 is a many- valued function, which increases by 2n-t when z describes a
closed path round the origin.
For put «=r (cos 6-\-imi 6).
Then one of the vcJues of log 2, on this definition, is
Limit n if* (cos — Hisin- |-lL
»=■« I \ n M/ , J '
1
where r* is the positive nth root of r.
This can be written
1
Limit n {r* - 1} + i6,
naoo
«
When z describes a closed path round the origin, the first term in this expression
remains unaltered, while the second increases by 2frt ; hence the result.
Example 2. Find the points at which the following functions are not regular.
{a) «*. Answer, «— 00.
(6) cosecs. Answer, z=0, ±ir, ±2ir, ±3ir, ....
z-\
^^^ «»-5«+6-
Answer, z^2,Z,
1
{d) e\
Answer, «=0.
(6) {(^-1)4*.
Answer, «=0, I, 00.
Example 3. Prove that if the different values of a», corresponding to a given value of z
are represented on an Argand diagram, the representative points will be the vertices of an
equiangular polygon inscribed in an equiangular spiral, the angle of the spiral being
independent of a.
(Cambridge Mathematical Tripos, Part I., 1899.)
47, 48] THE FUNDAMENTAL PEOPERTIES OF ANALYTIC FUNCTIONS. 69
47. LtouvUle's Theorem.
We know by § 38 that if /(^) be any function of z which is regular at
all points of the ir-plane within a circle G, of centre a and radius r, then
^ ^^^ 27^tjc(^-a)«+^•
Now let M be the greatest value of \f(z)\ at points on the circle 0.
Then this equation gives (§ 32)
nlM
From this inequality an important consequence can be deduced. Suppose
that /(z) is, if possible, a regular function of z over the whole z-plane,
including infinity, ie. that it has no singularities at all.
Then in the above equation M is finite when r is infinite, whatever n is ;
and therefore /<*^ (a) is zero for all values of n and a, i.e. /(a) is a constant
independent of a. We thus arrive at Liouville*8 theorem^ that the only
fwnction which is regular everywhere is a constant.
As will be seen in the next article, and again frequently in the latter half of this
yolnme, Liouville's theorem furnishes short and convenient proofis for some of the most
important results in Analysis.
48. Functions with no essential singularities.
We shall now shew that ^ ofnly one-valued fu/nctions which have no
singularities in either the finite or infinite part of the plane, except poles, are
roMonal functions.
For let / (z) be such a function ; let its singularities in the finite part
of the plane be at the points Ci, Cj, ... c*: and let the principal part (§44)
of its expansion at the pole Cy be
Z-'Cr (z^CrY (^-Cr)"^
Let the principal part of its expansion at the pole £: = oo be
if 5 = 00 is not a pole, but a regular point for the function, then the coefficients
in this expansion will be zero.
Now the function
^^^ r%\z-Cr^iz-Cry^-^{z-Crrr]
70 THE PROCESSES OF ANALYSIS. [CHAP. III.
has clearly no singularities at the points Oi* ^> ••• Cjb, x ; it has therefore no
singularities at all, and so by Liouville's theorem is a constant ; that is,
f{z) = constant 4- Ojir + (V* + ... + (ti^
f{z) is therefore a ratioual function, and the theorem is established.
Miscellaneous Examples.
1. Obtain the expanBion
2. Obtain the expansion
/(.)-/W+"i^°[/-W+/'(.)+!{/'(.+^)+/'(»+^)+-
+ .... (Corey.)
3. Obtain the expansion
+ ... (Corey.)
4. In order that values U-\- Vt, which are given as continuous functions of the arc
of a circle, should be the boimdary values of an analytic function, shew that it is necessary
and sufficient :
(a) That — '^ — ^^^ — — — -^ at the place ^=0 should be uniformly integrable for
all values of a ;
{h) That the values of 7 shall be given by
V{a)^^ i' {U{a'-^)'U{a^^)}Qot^d^. (Tauber.)
MISC. EXS.] THE FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS. 71
5. Shew that for the series
«i»0
2»+«-*'
the region of convergence consists of two distinct areas, namely outside and inside a circle
of radios unity, and that in each of these the series represents one function and represents
it completely.
(Weierstrass.)
6. Shew that
(Jaoobi & Scheibner.)
7. Shew that
+*"<"*+^^-;-^'"+**>(i-»)-'/'V(i-<)«'-t dt.
(Jacobi & Scheibner.)
'8. Shew that
(i-^-.;^.,.-«-.*.=i;{,.=±|.*....<:-:|tg*-;)^-.}
^c--'-' ,i:t>y.y/s.':'j :'--<'-'-'-''"-
(Jacobi & Scheibner.)
9. I( in the expansion of (a+Oi^+tv')*^ by the multinomial theorem, the remainder
after n terms be denoted by i^ so that
(a+ai«+a^«)«»=ilo+i4i2+ii22*+...+-4n-i«*"^+^
shew that
i2»(a+ai*+<V«)-j^ (^^^^^^pri <«•
(Jacobi & Scheibner.)
10. If (ao+ai«+a^)-*-* j{ao+a^t+a^t*)'^dt
be expanded in ascending powers of z in the form
shew that the remainder after (n— 1) terms is
(Jacobi & Scheibner.)
11. Shew that the series
2 {l + X,(.)e'}-^jr-,
nao
where X,»(«)= -l+^-gj + ^j -...±^,
72 THE PROCESSES OF ANALYSIS. [CHAP. III.
and where 4> (^) is a regular function of z near «aO, is convergent in the neighbourhood of
the point «=0; and shew that if the sum of the series be denoted bj /{$), then /(f)
satisfies the differential equation
/' {e) -/(«) - (z). (Pincherie.)
12. Shew that the arithmetic mean of the squares of the moduli of all the values of
the series 2 Oj^^ on ^ circle |f|»r, situated within its circle of convergence, is equal
to the sum of the squares of the moduli of the separate terms.
(Qutzmer.)
13. Shew that the series
00
converges when |f | < 1 ; and that the function which it represents can also be represented
when \z\ < 1 by the integral
/a\4 /■• $'» dx
\w/ Jo' ^-z X*'
and that it has no singularities except at the point f ~ 1. (Lerch.)
14. Shew that the series
2 2 ( z «-i 1
- (* + ^*) + - 2 l^j _ 2^_ 2^,^.^ (2v+ 2v'«)« ■*■ (1 - 2^- 2^z-H') (2p+2p'z'hyi '
in which the summation extends over all integral values of p, i/, except the combination
(iraaO, V »0), convergcs absolutely for all values of z except purely imaginary values ; and
that its sum is '+ 1 or - 1, according as^the real part of z is positive or negative.
(Weierstrass.)
15. Shew that sin •Ite ( « + -H can be expanded in a series of the type
Z Zr
in which the coefficient of either t^ or z'^ia
5-1 sin (2tt00s ^) cos nBdB,
16. If
hew that /(f) is finite and continuous for all real values of f, but cannot be expanded as
Maclaurin's series in ascending powers of z ; and explain this apparent anomaly.
CHAPTER IV.
Thb Uniform Convergence op Infinite Series.
49. Uniform Convergence.
We have seen* that the sum of a convergent series of analytic functions
of a variable z can have discontinuities as z variea It was found by Stokes "f
and Seidell in 1848 that this can never happen except in association with
another phenomenon, that of non-uniform convergence, which will now be
investigated.
Consider the series
« ?! . ^ + ^(2^-1)
( 1 + 2^) ( 1 + 2^ + ^2) "^ ( 1 + 2z + ^) ( 1 + 3^ + ^) "^ • • •
. 5 + g^U-l)
We shall first shew that this series is convergent for all values of z except
certain isolated points.
For, except for the roots of 1 + w-r + ^ = 0, the nth term can be put in
the form
1 1
l+n^ + -8^ l+(n + l)^H- sf^-^^ '
so the sum of the first n terms is
o 1 1
**- 1 + 2z i + (n + 1)« + -2*+* '
which, as n becomes infinitely great, tends to the value ^ ^ for all points
except 5 = 0: and for -r = 0, we have /S = 0.
Thus (except at the roots of the equations I + nr + ^ = 0) the series
converges ; and it represents a regular function, except at ^ = 0, where it has
a discontinuity.
* In § 42.
t Collected Paper$, Vol. i. p. 286.
t Miineh. Ahh.
74 THE PROCESSES OF ANALYSIS. [CHAP. IV.
What lies at the root of the discontinuity ?
The remainder after n terms is
For ordinary values of z, say ^ = 1, this remainder decreases rapidly as
n increases. Thus if w = 10, j? = 3, the remainder = o j. . on > * negligible
quantity. But now let z approach near to its discontinuity 0: say
^~ 1000000* "r^®^ ^i^'^ ^^^ value of z, the remainder after 1000 terms is
nearly 1, and the remainder after 1000000 terms is still nearly s- This
shews that, as z approaches the discontinuity at ^=»0, the terms which
contribute sensibly to the sum tend to recede to the infinitely distant part o/ the
series, so the first 1000 terms do not furnish a good approximation at all.
We can express this analytically as follows : — The number of terms n,
which we have to take in order to make |i2n| less than a given small positive
quantity e, tends to oo as we approach the point of discontinuity.
This circumstance is the basis of the following definition : —
Let Ui{z)-\- u^{z)'\-Ut{z)-\-u^{z)'\- ,..
be a series of functions of z, which is convergent at all points z within a given
area in the j?-plane. Let Rn be the remainder after n terms. Then since
the series converges, if we take a small finite quantity e we can find at any
point on the area a number r (varying from point to point) such that \Rn\ < €
so long as n > r. If the numbers r corresponding to the aggregate of points
in the vicinity of a given point z are all less than some definite finite number,
the series is said to be uniformly convergent at the point z ; but if near any
point z the number r tends to infinity, so that no definite upper limit can be
assigned to it, the convergence of the series is said to be non-uniform* in
the neighbourhood of the point z.
Example 1. Shew that the series
■*■ 1 +a* "^ (1 +««)«'^ •'• ■*"(TT?)* "*"'"• *
which converges absolutely for all real values of is, is discontinuous at z—O and is non-
uniformly oonvei*gent in the neighbourhood of ^— 0.
The svun of the first n terms is easily seen to be 1 +«*— ,-^ — sxr;:^ . So when z is not
zero the siun is 1 +^, and when z is zero the sum is zero.
* An interesting geometrical treatment of uniform convergence is given by Osgood in VoL lu.
of the BvU. of the Amer, Math, Soe, p. 59 (1896).
49] THE UNIFORM CONVEROENCE OF INFINITE SERIES. 75
The remainder after n terms is 7rTT2\»=i* "^^^ ^^^^ ^ made smaller than any
log-
aasigned smaU finite positive quantity c by choosing n so that n-l> j j m . But as
9 tends to aero, . 7|^r^\ tends to infinity, so n must tend to infinity, i.e. we have to
include an infinite number of terms in order to get the remainder less than c . This series
is therefore non-uniformly convergent in the neighbourhood of e=0.
Example 2. Shew that at «=0 the sum of the series
z z z
l(z+l) ' (a+l)(25+l)"-^{(n-l)«+l}{fw+l}
is discontinuous and the series is non-uniformly conveigent.
The sum of the first n terms is easily seen to be 1 ^ : so when z is zero the
sum is 0.
The remainder after n terms of the series is — — : ; so when z is nearly zero, say
fs one-hundred-millionth, the remainder after a million terms is --^^ or 1 - =^ , so
100^
the first million terms of the series do not contribute one per cent of the sum. And in
general if i be small, it is necessary to take n large compared with the large quantity
- in order to make the remainder after n terms smalL There is therefore non-uniform
z
convergence in the neighbourhood of 2=0.
Example 3. Discuss the series
»=i{l+»M{l-Kn+iy«««}-
The nth term can be written ; 5-^ - _— ^ Tr^— , , so the sum to infinity is r— — 5 ,
l-»-nV l-|-(n-|-l)'«* l-l-«*
and the remainder after n terms is z— ^ tth-, .
1-H(n-|-1)*2*
However great n may be, i^we take z equal to -tt> ^^^ remainder will have a finite
value, namely \ ; the series is therefore non-uniformly convergent at 2^0.
NoTB. In this example the sum of the series is not discontinuous at ««>0.
Cayley* regards non-uniform convei*gence as consisting essentially in the occurrence of
a discontinuity in the sum of a series. The condition for a discontinuity in a series
at the point z^av& that the series
T^\ a-z
shall have an indefinitely large sum when (a— z) is indefinitely small.
* ** Note on Uniform Oonvergeaoe," Proc, Hoy, 80c. Edinb. xix. (1891-2), pp. 208-8.
76 THE PROCESSES OF ANALYSia [CHAP. lY.
Thus in the series
(l-^)+«(l-;?)+z«(l-«)+...,
which is non-uniformly convergent and discontinuous at ^a*!, we have
a — z
= -«*, when a=l,
80 the sum of the series 2 ^^-^ is ; — , which is infinite for «= 1.
60. Connexion of discontinuity with non-uniform convergence.
We shall now shew that ths sum of a series of continuous functions of z,
if it is a uniformly convergent series for values of z within certain limits,
cannot he discontinuous for values of z within those limits.
For let the series be /(2r) = 1^1 (^) + 1^3 (<2:)+...+ 14^ (2r)+... = flfn(^) + lJnW>
where B^ is the remaiDder after n terms.
Since the series is uniformly convergent, we can to any small positive
number e find a corresponding integer n independent of z, such that
I iZn (-3^) I < « for all values of z within the area^.
Now n and € being thus fixed, we can, on account of the continuity of
Sn (z), find a positive number 17 such that, when |5 — / 1 < 17, the inequality
is satisfied.
We have then
<\Sn(z)''Sn(z')\ + \Rn(z)\'^\Itn{z')\
which establishes the fiesult.
Exam^ 1. Shew that at «=0 the series
1 1_
where Wj («)==«, «»(«)=«*•"*-«*•"•',
and real values of z are concerned, is discontinuous and non-uniformly convergent
1
The sum of the first n terms is i^~^ ; as n tends to infinity, this quantity tends to
1, 0, or - 1, according as 2 is positive, zero, or negativa The series is therefore absolutely
convergent for all values of z, and has a discontinuity at f =0.
1
The remainder after n terms, when z is small and positive, is 1 -i^'^ ; however great
n may be, by taking «=c~(*»~i) we can cause this remainder to take the valuie 1 - - , which
is different from zero. The series is therefore non-uniformly convergent at «>bO.
50, 51] THE UNIFORM CONVERGENCE OF INFINITE SERIES. 77
ExamjiU 2. Shew that at 2=0 the aeries
n-i{H-(l+^)*-i}{l +(!+«)*}
is discontinuous and non-uniformly conveigent.
The nth term can be written
l-(l+g)* _ l-.(l+g)*-i
l+("l+2)"* l+(l+z)»-i'
the sum of the first n terms is n . /i , w • Thus considering real values of t greater
than - 1, it is seen that the stun to infinity is 1, 0, or - 1, according as 2 is negative and
greater than -2, zero, or positive. There is thus a discontinuity at 2«>>0. This discon-
tinuity is explained by the fact that the series is non-uniformly convergent at «=0 ; for
the remainder after n terms in the series when t is positive is
-2
and however great n may be, by taking e=- this can be made to take the value
— 2
, which is difilerent from zero. The series is therefore non-uniformly convergent
so
1+e
at «=0.
61. Distinction between absolute and uniform convergence.
The uniform convergence of a series does not necessitate its absolute
convergence, nor conversely. Thus the series (§ 49, Ex. 1) S .^ ^.^ con-
verges absolutely, but (at ^ = 0) not uniformly : while if we take the series
« (, l)n-l
its series of moduli is
which is divergent, so the series is only semi-convergent ; but for all real
values of 2, the terms of the series are alternately positive and negative and
numerically decreasing, so the sum of the series lies between the sum of its
first n terms and of its first (n + 1) terms, and so the remainder after n terms
is less than the nth term. Thus we only need take a finite number of terms
in order to ensure that for all real values of z the remainder is less than any
assigned quantity, i.e. the series is uniformly convergent.
Absolutely convergent series behave like series with a finite number of
terms in that we can multiply them together and transpose their terms.
Uniformly convergent series behave like series with a finite number of
terms in that they are continuous and (as we shall see) can be integrated
term by term.
78 THE PROCESSES OF ANALYSIS. [CHAP. IV.
62. Condition /or uniform convergence,
A sufficient though not necessary condition for the uniform convergence
of a series may be enunciated as follows : —
If for all values of z within a certain region the moduli of the terms of a
series
8 = v^{z) -k- v^{z) '\- u^ W + ••'
are respectively less than the corresponding terms in a convergent series of
positive constants
then the series 8 is uniformly convergent in this region. This follows from
the fact that, the series T being convergent, it is always possible to choose n
so that the remainder after the first n terms of T, and therefore of 8, is less
than an assigned positive quantity € ; and since the value of n thus found
is independent of z, the series 8 is uniformly convergent.
GoroUary. The theorem is still true if the moduli of the terms of <S,
instead of being less than the terms of T, are to them in a variable but finite
ratio.
Example. The series
COSZ+gjCOS*;? + ^COS'« +...
is uniformly convergent for all real values of z, because the moduli of its terms are not
greater than the corresponding terms of the convergent series
1+^+1+
whose terms are positive constants.
63. Inteffration of infinite series.
We shall now shew that if 8(z)=^iii(z) + ti^{z)-\' ... is a uniformly con-
vergent series of continuous functions of z, for values of z contained within
some domain, then the series
jtifi(z)dz'\-jv^(z)dz + ... ,
where all the integrals are taken along with some path C in the domain, is
convergent, and has for sum l8(z) dz.
For let n be some definite finite number, and write
8 (z)=^ U,(Z) + U^(z)-¥ ... '\' Un(z) + Rn{z),
so
j 8(z)dz== jtii(z)dz+...'^ Iun(z)dz + lRn(z)dz.
52, 53] THE UNIFORM CONVEBGENCE OF INFINITE SERIES. 79
Now since the series is uniformly convergent, to every positive number €
there corresponds a number r independent of z, such that when n^ r we have
R^{z)\< €, for all values of z in the area considered.
Therefore if 2 be the length of the path of integration, we have (§ 32)
/'
<d.
Rn{z)dz
Therefore the modulus of the difference between I S(z)dz and the sum
of the n first terms of the series X I Un (z) dz is less than any positive
number provided n is large enough. This proves both that the series
2 jt^(z)dz is convergent, and that its sum is J8(z)dz,
Example 1. As an example of the necessity of this theorem, consider the series
• 2z{n(n + l)sin*«2- l}co8«'
2
««i {l+n''sin2«8}{l + (n+l)«8in««2}*
The nth term is
2zn cos 2* 2z(n+l) cos z*
l+n«sin2 2«"' ! + (« + !)* sin^z*'
and the sum of n terms is therefore
2gcosg' 2g(n+l)cosg*
The series is therefore absolutely convergent for all real values of ;; : but the remainder
after n terms is
2z{n+l)co8z'^
H.(n+l)2sin2?'
and if n be any number however infinitely great, by taking z= — -i this has the finite value
2. The series is therefore non-imiformly convergent at ;?«0.
Q* COS Z^
Now the sum to infinity of the series is i— .—5-,, and so the integral from to « of
the sum of the series is tan~^ (sin ^). On the other hand, the sum of the integrals from
to i? of the first n terms of the series is
tan~* (sin a^) - tan** (n-fl sin «*),
and foms= 00 this tends to
tan~*(sin«')— ^.
Therefore the integral of the sum of the series differs frx>m the sum of the integrals of
the terms by •= .
Example 2. Discuss the series
• 2g^{l-n(e-l)+e*'^^^}
,2in(n+l)(l+^«^)(l+e»+»««)
for real values of z.
80 THE PROCESSES OF ANALYSIS. [CHAP. IV.
The nth term of the series may be written
n(l+e^«) (n+l)(l +«*+»««)•
The sum of the first n terms is
1+^" (71+ 1)(1 +€•*!««)•
The series therefore converges to the value ^ ; and since the terms are real and
ultimately of the same sign, the convergence is absolute. The integral from to « of the
sum of the series is
log (1 +«?«).
The sum of the first n terms of the series formed by integrating the terms of the series
is
log(l+e0«)-^log(l+e»*ii^),
which fornsoo tends to
l0g(l+6««)-l
This discrepancy is accounted for by the non-uniform convergence of the series at f «0 ;
for the remainder after n terms in the original series is
or
(n+l)(l+«-^) «_+?,-t+(n+i);
i
and however great n may be, on taking «=> -— -r this takes the value unity ; so the series
is non-uniformly convergent at f ^0.
Example 3. Discuss the series
W1+W8+W3+...,
where
Uj ■= ze-'^f tt» = nw-"** - (n - 1) -w^*-*)**,
for real values of z.
The sum of the first n terms is nzer*^, so the sum to infinity is for all real values
of z. Since the terms Un are real and ultimately all of the same sign, the conveigenoe
is absolute.
In the series
/ riidz+l t^dz+ I i(^dz-\',„f
Jo Jo Jo
the sum of n terms is ^ (1 - 0"**^), and this tends to the limit ^ as n tends to infinity ; this
is not equal to the integral from to i? of the sum of the series 2 u^.
The explanation of this discrepancy is to be found in the non-uniformity of the
convergence neariS^O, for the remainder after n terms in the series Wj+Wj-I-... is -Twe-*^ ;
and however great n may be, by taking 2= - we can cause this to tend to the limit - 1,
which is different from zero: the series is therefore non-imiformly convergent near z=0.
54, 55] THE UNIFORM CONVERGENCE OF INFINITE SERIES. 81
64. Differentiatimi of infinite series.
The converse of the last theorem may be thus stated :
If 8 (z) ^ Ui(z) -h u^{z) + , .. is a convergent series of analytic functions of z,
which are regular when the variation of z is restricted to be within a certain
domainyandiftheseries'S.{z)^-r'V^{z)-\--j-ii>2(z) + ... is wniforrrdy convergent
within this domain, then this latter series represents -j- 8 (z).
For by the preceding result, if a and z are two points within the domain,
we have
r X(t)dt = r ui if) dt + \\i (t)dt+,..
J a J a J a
Since
if^(z) + u^(z)+ ,,. and i^i(a) + ti,(a) + ...
are each of them convergent series, we can write this
J c
^{t)dt = {i^i (z) + iLi{z) + .,.]-'{ui (a) + Wa(a) + ...}
= 8(z)^8(a),
and hence \
We may note that a derived series may be non-imiformly convergent even when the
original series is uniformly convergent : for instance the series
sin«— J8in2«+Jsin3«+..,
is non-uniformly convergent at z=^Tr; although the series from which it can be derived,
namely
- cos a+5jCOS 22- ^ cos 3«+ ...,
is uniformly convergent for all real values of z.
66. Uniform convergence of Power-8eries,
We shall now shew that a power-series is wniformly convergent at all
points within its circle of convergence.
For let jR be a region, forming part of the area of the circle, and let r be
a quantity greater than the modulus of every point of i2, but less than the
radius of convergence. Then if z be any point of R, the moduli of the terms
of the series
ao + aiZ + a^-^- ...
w. A. 6
82 THE PROCESSES OF ANALYSIS. [CHAP. IV.
are less than the moduli of the corresponding terms of the convergent series
ao + ctir + a,i'*+ ... .
But the latter series does not involve z, and so (§ 52) the power-series is
uniformly convergent within the region R\ as R is arbitrary, the series there-
fore converges uniformly at all points within the circle of convergence.
It must be observed that nothing is proved regarding points on the
circumference ; we do not even know that the series is convergent there at all.
Corollary, A power-series is continuous within its circle of convergence :
and the series obtained by differentiating and integrating it term by term
are equal to the derivate and integral of the function respectively.
Example, As an example of this, consider the series
which is convergent at all points within a circle of radius 1. We can integrate it term by
term, so long as the path of integration lies in this circle ; the result is
/;
Now / T-T-h clearly represents that value of tan~i» which lies between - ^ *^d +^ ,
So the series represents this value of tan" ^« and no other.
Miscellaneous Examples.
1. Shew that the series
represents . .^ when \z\<l and represents >, when |«|>1.
Is this fetct connected with the theory of uniform convergence ?
2. Shew that the series
2sin- +4sin;r +...+2*sin ;;— + ...
converges absolutely for all values of z, but does not convei^ uniformly near «bO.
00
3. If aserie8^(«)«r j (c^- c,,+j) sin (2i» + !)««• (in which Cq is zero) converges uni-
IT
formly in an interval, shew that g{z) -. is the derivate of the series
/(«)= 2 — sin2i«ir. (Lerch.)
CHAPTER V.
The Theory of Residues : Application to the
>
Evaluation of Keal Definite Integrals.
66. Residties.
If a point ^ = a is a pole of order m for a function /(z), we know by
Laurent's theorem that the expansion of the function near 2r = a is of the form
where ^ (s) is regular in the vicinity of ^ = a.
The coeflScient a»i in this expansion is called the residue of the function
f{z) relative to the pole a.
Consider now the value of the integral \f{z) dz, where the integration is
taken round a circle 7, whose centre is the point a and whose radius is a small
quantity p.
We have I f(z)dz=l, a^r f 7 ;z + / <l>{z) dz.
Now I <l> {z) rf^ = 0, since <f>(z)iB a regular function in the interior of the
circle 7 : and (putting z — a^ pe^) we have
, = p-»^-i r^ , when r + 1
= 0, when r^\.
But when r=! I we have
f J^ = ride = 2^.
jyZ — a Jo
Hence finally / f(z) dz = 2'rna^i.
Jy
Now let C be any contour, containing in the region interior to it a number
6—2
84 THE PROCESSES OF ANALYSIS. [CHAP. V.
of poles a, 6, c, ... of a function /(^), with residues a_i, 6_i, c_i, ... respec-
tively : and suppose that the function f{z) is regular at all points in the
interior of (7, except these poles.
Surround the points a, 6, c, ... by small circles a, /8, 7, ... : then since
the function f(z) is regular in the region bounded by (7, a, ^8, 7, ..., its
integral taken round the boundary of this region is zero. But this boundary
consists of the contour (7, described in the positive sense, and the contours
•1 A 7> ••• described in the negative sense.
Hence = [ f{z) dz -| f{z) dz-^j f{z) dz..,,
or ^ ~ I f{^)^^ ■" 27rwi«i — 27rt6_i ••• •
J c
Thus we have the theorem of residues, namely
f{z)dz^'l7ntR,
L
C
where 2-B denotes the sum of the residues of the function /{z) relative to
those of its poles which are situated within the coo tour 0.
This is an extension of the theorem of Chapter III. § 36.
67. Evaluation of real definite integrals.
A large number of real definite integrals can be evaluated by the use of
contour-integrals and the theorem of residues. The following examples will
serve to illustrate the various ways in which these aids to the evaluation
may be applied.
Example 1. To find the values of
P%«**co8(n^-sm^)cW and (^ e"^^ sin {n3- am 6) dS,
Denoting these integrals respectively by / and •/, we have
Write 0^=2, and let (7 be a circle of radius unity round the origin in the f -plana Then
«8 6 assumes the sequence of real values from to 2n-, 2; describes the circle C,
Hence l-iJ^-. I eFz'^'-^dz
^ J c
=s 27r X the residue of -—ry at «=0
_2.r
1
57] THE THEORY OF RESIDUES. 85
Therefore -^■" -r i
Example 2. The method used in Example 1 can be very generally applied to
trigonometrical integrals taken between the limits and 2v, As another example,
consider the int^pral
/-r f-, (a>6).
Write efi^^z'y and let C be the circle on the z-plane whose centre is at the origin and
whose radius is unity.
/ [ 2cb
Then
dz
» Jc
=4ir X simi of residues of j-^ — r. at poles contained within C.
Now
fea»+2€W+6 2^a^-b*-
-1
a V a* — 6* a v a* — 6*
Thtt^ore the two poles are at 2= - ^ r^- and f ■■ - ^ + ^ ^ ~ , and the residue
at the former (which is the only one within C) is -- — — .
Hence /«
Example 3. Shew that
(a+6cos5)2~(a2-'62)>/2'
Example 4. Find the value of
i:
/
* x sin ma? ,
ax.
-00 ^+a*
Let (7 be a contour formed by the real ckxis together with a semicircle y, consisting of
that half of a circle, whose centre is at the origin and whose radius is very large, which
lies above the real axis.
Then ^ - j is a function of z which has only one pole in the interior of C, namely at
,-— , dz = 2*rt X residue of -, ^ at its pole at. But writing
z^at-^-^y we have
^— ma
=-o-v + positive powers of f.
86 THE PROCESSES OF ANALYSIS. [CHAP. V.
Therefore ^ ^ = -y^—-ir+ positive powers of («-ai).
Thus the residue of -5- — = at at is - «"*•.
The«fo« .ir~-/^ -;^,«fc=(/^^ +/J /g,^
Since
is infinitesimal compared with pr at points on y, the integral round y is
infinitesimal compared with / —
Therefore wte"*
Equating imaginary parts, we have
or 2fr, and is therefore zero.
/* iT sin mx , _^
Example 6. To find the value of
/ ««~"**sin(asin6^)^— ^.
Take a contour C composed as in Example 4 of an infinite semicircle y and the real
axis.
Then / — .«<" ^ . ^ cfe—2»ri x residue of 77-. «** « . • at its poles inside C
But r\««« , A has only one pole in the interior of C, namely at the point z^rL
Now if 2«an+f, we have
2i^ ?+7i = 2i'^ 2nf+T* = 4^'^ + positive powers off.
1 6r
Therefore the residue is t-^' '
But at points on y, e^«*=0, so e<*« =1, and so
Jy2i a^+f^ 2tjy^"2*
Therefore ^eo^-^^^^./" e^ooBhx^j^^amibx) ^^,
or JVoo.ta8m(a8in6*)J;^=^ (««-»'-!).
We may note that in the above 1 stands for the limit of 1 where k is infinitely
/■*
great, and is not equal to the limit of 1 where k and I are different.
EXS.] THE THEORY OF RESIDUES. 87
Example 6. Prove by integrating
/:
^dz
round the contour used in Examples 4 and 5, that
Example 7^ 'Find the value of
/,
* mnmx .
Consider a contour C, formed of 1<> a semicircle r whose centre is at the origin and whose
radius is very large, 2^ a semicircle y whose centre is also at the origin and whose radius is
very small, and 3** the portions of the real axis intercepted between these circles. The semi«
circles are to be drawn in the upper half of the «-plane, i.e. the half above the real axis.
. , . — 5rs«2frt X the residue of -7-,-:— -v at the singularity
Zmeat.
But if we write 2= at + (> where C is small, we have
Thus the residue at at is -
Therefore -
4a»
(-!)•
ia?
(**+a)' jc «(«»+««)« ~U-. "^jr JvJ*(»«+a«)»-
Now / a . jNg is infinitely small at points on r, so the integral taken round r vanishes.
_co«(2:«+a*)* a* 20* \ a/
In this, I means f + f > where the two c's are the same : but in the final result
J -00 J 9 J —00
we can put f >bO, since the final integrand is finite at the origin.
Equating the imaginary parts on both sides of this equation, we obtain
/* sin mxdx w irg"*^ / 2\
-ooi"(^+a«)«"^" 2a» [^'^aj'
J /"• Binmxdx it ire"^^/ . 2\
And so I —7-5 oTo^TT-i 7— r— |w*+-|.
Jo^{x^-^ay 2a* 4a» \ a/
Example 8. Find the value of
/:.
* cos 2ax - cos 26^ ,
1^ '^-
Take the contour C formed as in Example 7 by an infinite semicircle r, a small semi-
88 THE PROCESSES OF ANALYSIS. [CHAP. V.
circle y round the origin, and the parts of the real axis intercepted between them. Within
this contour the function — |- has no singularities.
In this equation i must be r^arded as an abbreviation for i + i where f is the
radius of y.
^8«<* 1
Now at points on r, —^ is ssero compared with - , so the integral round V is zero.
z* " z
-— cb a one-half of 2tnx the residue of — ,- at the origin
am X the residue of =
= — 2ira*
—^ dz=-27ray
— 00 ^
^ cfe=2fr(6-a).
Taking the real part of this we have
/* cos 2a:r - cos 26^ j ^ ,, \
^ flte=2»r(6-a),
, . cos 2cu? - cos 2&r . i, .^ i ^ j i _j__- x T* ^
and smoe -^ is nmte when ^=0, we need no longer restnct / to mean
Example 9. Find the value of
/o'--«^(T-^)^f^ («>0).
We have J^ ^-i sin (^ - 6*) ^^
Consider a contour (7, formed as in Examples 7 and 8 by an infinite semicircle r,
a small semicircle y round the origin, and the parts of the real axis intercepted between
them.
Then i j (-«y»-ic««^^=2irtxthe residue of | (-^O^'^^^^T^ft^^ i^ singularity
«=ir.
EXS.] THE THEORY OF RESIDUES. 89
Putting 2arn+C <^^ n^ecting powers of (y we see that the expansion of
begins with a term
"4— f— '
80 the required residue is — j r*-* c-^.
Therefore s^"*«"^=5 / (-^t)"-^
z z J c
rdz
At points on r the integrand is infinitesimal compared with - , and so the integral
z
round r is zero.
At points on y the integrand is approximately ^—^ — z^^ and so if a > the integral
round y is zero.
Therefore / ^^s- (^-fe-)^-g, = | /^J-^T-.-
/* /it:
gaooBtegin (a sin 6^) — .
^
Wehave f Voo«to8in(asin6x)~ =i f * «<»^— ,
Jo 'a; 2i j_ao 4? '
where in the latter int^ral i must be regarded as an abbreviation for i +1 where
€ is a small quantity.
Take a contour C, consisting as in Examples 7, 8, 9, of an infinite semicirole r, a small
semicircle y of radius t roimd the origin, and the parts of the real axis intercepted between
them.
Then 0= [ e^^^ [ eo.^^- ( ea^^+ T e^"^ .
J C X JT ^ Jy ^ J '• ^
At points on r, we have 6*«*=0, tf<»<^=l, and so
J T X JTX
At points on y, 6***= I, so
Jy ^ Jy^
Therefore [" ga«fc»<?ff =^(c«-i),
/oe J
^oo«6xgin(asin&ar) - =3(e«-l).
round the same contour as that used in Examples
7, 8, 9, 10, shew that ('^?^^cLv=Z.
jo ^ 2
90 THE PROCESSES OF ANALYSIS. [CHAP. V.
Example 12. To find the value of
I :r-— cKr, and/ , dx (0<a<l).
Jo 1+^ jo 1-^
Write /=/ f— dar,andfi'-/ f—dx.
Jo l+a? J Q I -X
Ab will be seen from the working below, the integral K has a meaning only when I is
+ I , where / is a small positive quantity,
i-f^ Jo
Consider a contour C formed of (a) that half r of a circle, whose centre is at the origin
and whose radius is a large quantity 72, which is above the real axis, (b) that half y of a
circle whose centre is at the origin and whose radius is a small quantity r, which is above
the real axis, (c) that half y' of a circle, whose centre is at the point ( - 1) and whone
radius is a small quantity r', which is above the real axis, (d) the parts of the real axis
intercepted between these semicircles.
— , where the many- valued function is supposed to have that one of
its determinations which is real and positive when z is real and positive. The integrand is
regular in the interior of the contour (7, and so
f 2f-^dz
Now on y the integrand is sensibly equal to 2i^'\ and so the integral to — L which
-r
is infinitesimal, since a > 0.
(-1)0-1
On y, the integrand is sensibly equal to ^ J^ — ; putting 1 + « «■ r'tf*', the integral
1 "rZ
along y' is / %d$^ or iV ( - 1)«~*.
On r, the integrand is sensibly equal to -^^ , the modulus of which is infinitesimal
m
compared with , -: ; so the integral along F is zero.
Thus iri=( -!)*-« /+Jr=-/(cosair-tsincwr) + ir.
Therefore equating real and imaginary parts, we have
sin air '
K= IT cot air.
Example 13. By using the result
f'^ x'-^dx _
Jo l+x ~sL
sm atr
shew that r-^==^—. Limit 2 -r — . (Kronecker.)
68] THE THEORY OF RESIDUES. 91
68. Evaluation of the definite integral of a rational function.
The principles which have been applied in the preceding paragraph can
also be used to evaluate an integral of the form
f f{!c)dx,
J —00
where /(a?) is a rational function of x, in the cases when this integral has a
meaning.
a ioD\
For suppose that f{a) is brought to the form of a quotient Yj-i , where
g (x) and h (x) are polynomials in x. In order that the integral may have a
meaning unconditionally, it is necessary that the degree of g (x) should be
at least two units lower than that of h (x), and that the equation h(x) =
should have no real roots.
Consider now a contour C, formed of the real axis together with a
semicircle F of large radius, whose centre is at the origin, and which lies in
the upper half of the z-flsLne.
We have 1 f{z)dz — 2iri x sum of residues oi f{z) at the poles oi f{z)
J c
contained within C,
Now / = I + / • *^^ since f{z) has a zero of at least order 2 at
jt = 00 , it follows that / is zero.
Hence I f{x)dx^ 27n x sum of residues of f{x) at those of its poles
/ —00
which are contained in the upper half of the ^-plane.
If the d^;ree of g (x) is lower than that of A (x) by only one degree, or if A (x) has real non-
repeated roots, the integral will still have a meaning provided we make certain restrictions,
Le. that I shall be imderstood to mean the limit, when k tends to oo and t to zero^ of
where c is a typical root of the equation A (:7) =0.
J'-^L'
Example I. The function TyTTSs ^^^ ^ single pole in the upper half of the 2-plane,
3i
namely at 2»t, and the residue there is -tt.', we have therefore
/:
dx 3ir
(^+1)3 8 *
Example 2. Shew that / 7 — — , jr-. = 5-1 •
92 THE PROCESSES OF ANALYSIS. [CHAP. V.
69. GaiLchys integrals
We shall next discuss a class of contour-integrals which are very fre-
quently found useful in analytical investigations.
Let (7 be a contour in the ^-plane, and let f{z) be a function regular
everywhere in the interior of C, Let <^ {z) be another function, which in the
interior of C has no singularities except poles ; let the zeros of <^ {z) in the
interior of C be Oi, Oj, ..., and let their degrees of multiplicity be rj, r„ ...;
and let its poles in the interior of (7 be 6i, 6j, ..., and let their degrees of
multiplicity be «i, 5a, ....
Then by the fundamental theorem on residues, we have
^ — ; I f(z) %-T-l dz = sum of residues of -4^-}—- in the interior of C.
2'inJc <t>{^) <f>(^)
Now !T^ c*^ bave singularities only at the poles and zeros of
<f> (z). At one of the zeros, say Oi, we have
<f>{z) = A(z- OiY' +B(z-aiy^+' + ....
Therefore <^' (z) = Ar^ (z - o,)'"*-^ + B(n + l){Z''a,y^-{- ...,
and /(z) =/(a,) + (^ - Oi)/' (aj) + . . . .
Therefore !f^ = + a' constant + positive powers of (z — Oi).
<P yZ) Z — Oi
Thus the residue of !T} > at the point ^ = Oi , is r^f{a^.
Similarly the residue at -er = 6i is — «i/(6i) ; for near «r =s 6^, we have
and f{z) =/{bO + (-? - 6,)/' (bi) + • • ^
f(z)6' (z) -5,/(6i) . . r r
so "^ ^ , ; . a= I + a constant + positive powers of jp — 6i.
<f>{z) z-bi ^ ^
' Hence ^ f^/{z) *'|^J dz^t rj(a,) - 2 .,/(60,
the summations being extended over all the zeros and poles of <f> (z),
60. The number of roots of an equation contained within a contour.
The result of the preceding paragraph can be at once applied to find the
number of roots of an equation <f>{z) = contained within a contour C,
For on putting f{z) = 1 in the preceding result, we obtain the result that
^T— ^ I ^"7 \ dz is equal to the excess of the number of zeros over the number
2mJc<f>{^)
59 — 61] THE THEORY OF RESIDUES. 93
of poles of if>(z) contained in the interior of (7, each pole and zero being
reckoned according to its degree of multiplicity.
Example 1. Shew that a polynomial (t) of degree m has m roots.
Let <^(«) = ao2^+ai«^"* + ...+am-
Then ^ = ^og;r^^ >"-+^- ^ .
"" _ * y2' " "
For large values of «, this can be expanded in the form
*'W = ^ + ^
<l>{z) z^z^
Thus if C be a large circle whose centre is at the origin, we have
Hence as ^ (z) has no poles in the interior of C, we have
number of zeros of d> (z) = ^j— : j %-rl cfe"
m.
Example 2. If at all points of a contour C the inequality
is satisfied, then the contour contains k roots of the equation
For write /(«)=a^a!~+a„»--i«^~^+...+ai«+aQ.
Then /(«)=«».» A^ a^^-»+...+«.,, j^+ a^.^-'+...+a, ^
=a*«* (1 + i7) say, where | CT"! < 1 on the contour.
Therefore the niunber of roots of f{z) contained in C
2irt ; c /W
"2irijcV« 1 + C^ rfW
— =27rt ; and since |^| < 1 we can expand (1+ (7)~* in the form
Therefore the number of roots contained in C is equal to k.
61. Connexion between the zeros of a function and the zeros ofUe derivate,
Macdonald* has shewn that if f(z) be a regular function of z in the interior of a
contour C, defined by an equation \f{z)\=M where M is a constant, then the number of zeros
of f(z) in this region exceeds the number of zeros of the derived function f'{z) in the same
region by unity,
* Proc. Land, Math. Soc, xzix. (1898).
94 THE PROCESSES OF ANALYSIS. [CHAP. V,
For since f{i) has no essential singularity in the region, the number N of its zeros in
the region is finite. Now if m be a small number, the part of the locus |/(«)|=m in the
interior of the contour C consists of N closed curves surrounding the N zeros of f{z). As
m increases, these ovals increase, until two of them coalesce, the point at which they
coalesce being a node on the curve corresponding to that particular value of m. When
m has increased to its final value if, the N closed curves have coalesced into one closed
curve, and therefore N^-l nodes have been passed through. Each of these nodes is
a zero of f (z); for if /(«)=0+«V^, where and yfr are functions of x and y with real
coefficients, then ^ and -^ vanish at a node on the curve ^+^= constant; that is,
f'{z) vanishes. Moreover, two ovals cannot coalesce at more than one point, b» f{z) is
single-valued.
Hence the number of zeros of /' (z) inside the contour is (^- 1).
The proof assumes the zeros of f(z) in the interior of (7 to be all simple : the case where
f{z) has multiple zeros can be at once reduced to this, by dividing out the factor common
to f{z) €Uid /' (z). If /' (z) has two zeros equal, two of the double points coalesce, that is,
three ovals coalesce at the same point.
Similarly it can be shewn that the number of zeros of /' {z) in the region between the
contours |/(«)|«=mi and |/(«)|"««ij is equal to the number of zeros of f(z) in the same
region, if /(«) is regular in the region.
Example 1. Deduce from Macdonald's result the theorem that a polynomial of degree
n has n zeros.
Example 2. Deduce from Macdonold's result that if a ftmction /(z), regular for real
finite values of z, has all its coefficients real, and all its zeros real and different, then
between two consecutive zeros of f(z) there is one zero and one only of f'{z).
Miscellaneous Examples.
1. a function 0(«) is zero for «=0 and regular when I^Kl. If /(x, y) is the
coefficient of » in <^ (^+yO> prove that
'o i^2a>>cos^-h ^ -^(^^> sm^)cW-W>W.
(Trinity College Examination, 1898.)
« «i. XI- X r* sinew; , 1 e«+l 1 /t j v
2. Shew that j^ -__<ir-j^y-y--. (Legendre.)
3. By integrating I e-^dz round the perimeter of a rectangle of which one side is the
real axis and another side is parallel to the real axis and at a distance a from it, shew that
/:.
e-'*cos ^aidt^sfne-^\
'00
and I e-^ Bin 2(Udt^0,
/:
oi- XL X [ l-rco8 2^ , . ^j^ w, \-r
4. Shew that / , — ^^-r^ log sm 6dB = t log —j-
jo l-2rcos2^+r* ^ 4 ® 4
MISC. EXS.] THE THEORY OF RESIDUES. 95
6. Shew that
/,
a?cwr=--log(l+a) if -.l<a<l
l-2aco6a?+a* 4a
and = ^ log ^1 + ^^ if a« > 1. (Cauchy.)
6. Shew that
/,
00
sin 01^ sin (bqX sin d>-^ sin o^ , ir . .
• ••
if a be different from zero and
(Stermer.)
7. If a point z describes a circle C of centre a, any one- valued function u^f{z) will
describe a closed curve y in the t«-plane. Shew that if to each element of y be attributed
a mass proportional to the corresponding element of (7, the centre of gravity of y is the
f{z)
point r, where r is the sum of the residues of -^^-^ at poles in the interior of C.
z — a
(Amigues.)
8. Shew that
dx 7r(2q-hft)
i-«(^
-00 (^+ 6^) (^ + a2)« 2a26 {a-^hf '
9. Shew that
dx _ TT 1 .3. ..(2^-3) 1
/
»oo(a+6j?2)* 2*-»6* 1.2...(n-l) a^-h'
10. If /VW=(l-^)(l-47«)...(l-a;*-i)...(l-ar«)(l--^)... (1-;f«*-%.
... (l-a?»-i)(l-:c«»-«)... (l-x(»»-»>"),
shew that the series
convergee when »■ is not a root of one of the equations
1-0;
©■-
and that the sum of the residues of f{x) contained in the ring-shaped space included
between two circles whose centres are at the origin, one having a small radius and the
other having a radius between n and n-|-l) is equal to the number of prime numbers less
thann-hl.
(Laurent.)
CHAPTER VL
The Expansion of Functions in Infinite Series.
62. Darbov^'s formula.
Darboux has given* a formula from which a large number of expansions
in infinite series can be derived
Let f{z) be an analytic function of z, regulai* at all points z within a
circle of centre a and radius r; and let z he a, point within this circle.
Let (z) be any polynomial in z, of degree w. Then if R^ denotes the
expression
(- 1)« {z - a)*»+' C <f> (0/<"-^*' {a + t{z^ a)] dt,
Jo
where the integration is taken along the real axis of t, we have on integration
by parts
R^ = [\- ir{z- aY <f> {t)f^^ {a + e (^ - a)}J
+ (- 1)«-^ (z - a)« f'f (O/^*** {a + t(z-- a)} dt,
J Q
or i2n = (- lY (^ - ar {<t> (!)/<"» (z) - <t> (0)/<«» (a)}
+ (- D^^i {z - aY fV' (0/*"^ [a + t(z^ a)} dt
Jo
Integrating the last integral by parts in the same way, we obtain
i2„= (- ir (z - ar {<!> (l)/<~> W - <t> (0)/<«) (a)}
+ (- 1)~-» (^ - a)*^^ {f (I)/<'^»> (-?) - f (0)/(«-« (a)) + ...
- (^ - a) (<^<«-^) (1) r (^) - <^'^^^ (0)/' (a)}
+ (-e - a) f <^ w (0/' [a+tiz"- a)] dt.
Jo
Now <^<*> (t) is a constant independent of t, since <^ (t) is a polynomial of
order n ; and hence
(z - a) f ' <^^«) (0/' {a + < (^ - a)} dt = <^» (0) {/(z) -/(a)}.
♦ Liouville'8 Journal (3), u. (1876), p. 271.
62, 68] THE EXPANSION OP FUNCTIONS IN INFINITE SERIES. 97
Thus finally we have Darboux's formula
<f>^^ (0) {f(z) -/(a)} ^(z^a) {<^*-« (1)/' (z) - <^^-« (0)/' (a)]...
+ (- 1)» (iT - a)^' r (f> (t)/^"^'^ {a + t(z- a)} dt
Jo
Taylor's expansion may be derived from this formula by putting
^(^)aB(t — 1)", and then making n tend to infinity: other new expansions
may be obtained by substituting special polynomials of degree n for <f> (t), and
in the resulting formula making n tend to infinity : in each case it must
of course be shewn that Rn tends to zero as n tends to infinity.
Example, By subetituting 2n for n in Darbouz's formula, and taking it>{t)=:t^(t- 1)%
obtain the expansion
/(«)-/(«)- J^ ^"^^ynr''^V <"W+(-l)'"-V<"(a)},
and find the expreesion for the remaioder after n terms in this series.
63. The BemoulUan numbers and the Bemotdlian polynomials.
• z z
If the constants which occur in the expansion of ^ cot ^ fin ascending
powers of jt be denoted by J3i, jBs, J3t» ... > so that
z z s^ * s^ ^
then Bn is called the nth BemoulUan number. It is found that
-''1 — g I -^^a — 80 » •'^t = 42 > • • • •
The BemoulUan numbers can be expressed as definite integrals in the
following way.
TTT 1 r* sin ©a? da? S f* . ,
We have I J; , = 2 / €r^^ sm pxdx
00
= 2 — *^ —
= -1+1
Equating coefficients of j)*'*~' on the two sides of this equation, and writing
a = 2t, we obtain
A proof of this result, depending on contour integration, is given by Carda, MoncUthefte
fwr Math, und Phys. v. (1894), pp. 321-4
W. A. ' 7
98 THE PROCESSES OF ANALYSIS. [CHAP. VI.
BxampU. Shew that
* ir**(2»»-l)yo "smhF*
The Bemoullian polynomial of order n is defined to be the coefficient of
t* e** — 1 .
—. in the expansion of t -^rZTT ^ ascending powers of t It is denoted by
<f>n (z), 80 that
eV-T= 2 ^V (!)•
6*— 1 »■! n! ^ -^
This function possesses several important properties. Writing (z + 1)
for z in the preceding equation and taking the difiference of the two results,
we have
n-1 Tli
On equating coefficients of t^ on both sides of this equation we obtain
nz'^^ = <f>n(z+l)-<f>n(z\
which is a difference-equation satisfied by the function ^^ (z).
The explicit expression of the Bemoullian polynomials can be obtained
as follows. We have
and
e*-
t
tef + 1 t
«*-
l~2c«-l 2
~2 £ 4 2
= 2i~*2i-2
"^"2"^ 21 ~ 4! ■^••••
Hence
„r,~in r"^ 2r"^"3r'*'-|f "2■'"■2^"^■^••r
From this, by equating coefficients of <*, we have
the last term being that in ^ or ^ ; this is the explicit expression of the nth
Bemoullian polynomial.
mm
64] THE EXPANSION OF FUNCTIONS IN 'INFINITE SERIES. 99
The BemouUian numbers and polTnomials were introduced into analysis by Jacob
Bernoulli in 1713.
Example. Shew that
*nW=(-ir*i»(i-«).
64. The Maclaurin-Bernoullian expansion.
In Darboux's formula write <^ {t) = ^n (0> where 4>n (0 '^ ^^^ ^^^^ Bemoul-
lian polynomial
Now from the equation
<^(^+l)-.<^„(0 = n^-S
we have by differentiating k times
<^n«*»(e + l)-i^n^(0 = w(n-l)...(n-A)r-*-^
Putting ^ s in this, we have
But the value of <^n^' (0) is obtained by comparing the expansion
<^(^) = 0n(O) + ^<^,'(O)+|j<^"(O) + ...
with the expansion
Substituting the values of <^„**(1) and ^n**(0) thus obtained in Darboux's
result, we find what is known as the Ma^^laurin-Bernoullianformtda,
(z - a)/' (a) ^f{z) -/{a) - —^ {/' (,z) -/' (a)}
+ ^i%P^{/"(^)-/"(a)l + -.
•^ ^" ^^'(In"- 2)7 "^ t/*^ (^) -/"^ («)}
In certain cases the last term tends to zero as n tends to infinity, and we
can thus derive an infinite series from the formula.
Example, lff{z) be an odd function of «, shew that
JV*.(<)/«***"(-«+2«)<ft,
2nl
where ^. (t) is the Bemoullian polynomial of order n.
7—2
/
100 THE PROCESSES OF ANALYSIS. [CHAP. VI.
66. Burmann's theorem.
We shall next consider a number of theorems which have for their object
ths expansion of one function in powers of another function.
Let <f> (z) be a function of z, which takes the value b when z takes the
value a, so that
b = <t> (a).
Suppose that ^ (z) is an analytic function of z^ regular in the neighbour-
hood of the value -e? = a, and that (f/ (a) is not zero. Then Taylor's theorem
furnishes the expansion
<l>iz)-b = 4>'(a)(z-a) + ^^iz-ay+...,
and on reversing this series we obtain
which expresses -^ as a regular function of the variable {<f>{z) — 6}, for values
of z in the neighbourhood of a. If then f(z) be a regular function of ^ in
the neighbourhood of a, it follows therefore that /(-«?) is a regular function of
{<l> (z) — 6} in this neighbourhood, and so an expansion of the form
A') =/(«) + «! {* (^) - 6} + Jj {4> (^) - b}*
will exist, which, as it is a power-series in {^ (z) — 6}, will be valid so long as
\^(z)-b\<r,
where r is some constant.
The actual expansion is given by the following theorem, which is
generally known as Burmann's theorem.
If yfr (z) be a function of z defined by the equation
, . z — a
then the function f(z) can for a certain domain of valines of z be ea^pa/nded in
the form
fiz) =/(a) + 1^ ^^^^i,"^^" ^. [/' (a) tt («)}-] ;
and the remainder after n terms in the series is
i_ c' f r <i,(z)-b -\'-^f'(t )<f>'(z)dtdz
2',rijjyl4>{t)-bj if>(t)-<l>(z)
where y is a simple contour in the t-plane, enclosing the point < = a.
66] THE EXPANSION OP FUNCTIONS IN INFINITE SERIES. 101
To prove this, we have
- A. ['[ /' (0 <t>' (^) dt dz r ^(z)-h ^
iinJJy <f>{t)-b I TW^
{^(^)-fel-' -l
But —['[ \^(_')-i'\ V(t)<l>'(')cUdz {»(^)-6}*-» f /'(t)dt
ZmJaJyl'l>(t)-bj (f>{t)-b 2in{k+l) Jy{<t>it)-b}'^'
2^- (A + 1) J^ (< _ a)*+' - 2,ri(A+l)! ^* ^-^ («> t^ ^">5 J'
Therefore /(^)=/(a)/T^ ^^ii^* £.[/'(«) ftW]
Exampli
Lf'f lt(ft:lT~'/l(tl£Mdtdz
'^JaJyl<l>{t)-bj <f>{t)-<f>(2) •
where
27rt
To obtain this expansion, write
in the above expression of Burmann's theore^n ; we thus have
Zi
But
(&J '"^•'"•^}« -{^1 ''''^"'^),,, (P^**i^« *=«+')
=(n-l) ! X coefficient of ^-i in the expansion of tf-«^(>«+<)
=(n-I)!xooefficientof^-iin i (-l)^^^<^(g«+<r
r-o rl
^ ^ A(n-l-r)!(2r-n+l)r
The highest value of r which gives a term in the summation is r=n- 1. Arranging
therefore the summation in descending indices r, beginning with r«n- 1, we have
=(-i)«-ic;.
which gives the required result
«•«
ip
102
THE PROCESSES OF ANALTSia
[chap. VI.
Excmiple 2. Obtain the expression
««-sin««+- . 2®^ '+3~5 • 3 8m««+ ....
Example 3. Let a line p be drawn through the origin in the ;s-plane, perpendicular to
the line which joins the origin to any point a. It zh% any point on the «-plane which is
on the same side of the line p as the point a is, shew that
00 1 /«—a\ '"*■•' ^
66. Teixeira's extended form of Bunnann's theorem.
In the last paragraph we have not investigated closely the conditions of
convergence of Burmann's series, for the reason that the theorem itself will
next be stated in a much more general form, which bears the same relation
to the theorem just given that Laurent's theorem bears to Taylors series:
viz., in the last paragraph we were concerned only with the expansion of a
function in positive powers of another function, whereas we shall now discuss
the expansion of a function in positive and negative powers of the second
function.
The general statement of the theorem is due to Teixeira*, whose exposi-
tion we shall follow in the next two paragraphs.
Suppose (1) that/(^) is a regular function of ^ in a ring-shaped region Ay
boundid by an outer curve S and an inner curve s; (2) that 0{z) ia a
regular function everywhere inside /S, and has a single zero a within this
contour; (3) that x is the affix of some point within A; (4) that for all
points of the contour S we have
l^(a;)|< 1^(^)1,
and for all points of the contour s we have
i^(a;)i>i^(^)i.
The equation
has, in this case, a single root z=»x in. the interior of S, as is seen from the
equation
1 r e'(z)dz _ I [[ ff{z) , ^ ., . r ff{z) i
^ J_ r ff(z)dz
^27riJs 0{z) '
of which the left-hand and right-hand members represent respectively the
number of roots of the equation considered and that of the roots of the
equation 6{z)rsQ contained within 8.
* CreUe'9 Journal, cxxn. (1900), pp. 97—123.
mmm
66, 67] THE EXPANSION OF FUNCTIONS IN INFINITE SERIES. 103
Cauchy's theorem therefore gives
/M = J_ [( A^W('')ds [ Az)0'{z)dz -\
^^*^ 2%'7rlJae{z)-e(x) J, '^(z)-0(x)]•
The integrals in this formula can, as in Laurent's theorem, be expanded
in powers of 0(ai), by the formulae
f f{z)ff{z)dz _lff,,.t Az)ff{z)dz
We thus have the formula
where a ^ JL { fMli^l^
Sn=i^\nz)e^^{z)ff{z)dz.
This gives a development oi f{x) according to positive and negative
powers of 6 {x\ valid for all points x within the ring-shaped space A.
67. Evaluation of the coefficients.
If the function /(^) has no singularities but poles in the region limited
by the curve a, the integrals which occur in the preceding formula can
be evaluated in the following way.
Let 6i, &9, ... &A be the poles; and let Ci, C2, ... c*, c, be circles with
centres &i, 6s, ... 6;^, a, respectively, and with very small radiL
Then A - ^ ( A^)^(')d z_ 1 f /'(z)dz
"'iiriJs 0^' {z) ~ liri } s ne^ (,zj
= S JL/" f'i^ldz 1 (f'{z)dz
and Bn'^'-^j f{z)e^'{z)ff(z)dz
a?— a
Ix^a
104 THE PROCBSSBS OF ANALTS1& [CHAP. VI.
Thus if Om be the degree of multiplicity of the pole 6,», and if ^ ^ be
denoted by 0i (x), we have
" n!L<iB»-'K»(«)jJ,
+ I 1 r<^- y'(a>)(a>-6m)-^M '|
It may happen that a is also a pole of /(x). It is easily seen that in
this case A^, is given by the formula
■^ (n + /9) ! n [dafi+^ \ ^,» (a;]
where /3 is the degree of multiplicity of the pole a ; the formula for Bn must
likewise be replaced by
- ^inoni [^ ^f <*> ^'" <-) <- - '^>'"-"j].^'
when w ^ )8.
The preceding formulae do not give the value of Aq ; this can be found
from the formula
A ^ i l_f f{^)^i^)dz . 1 { f{z)ff{z)dz
"^'"jLiiiTrJc^ 0iz) ^iiirj, 0(z) '
which gives
when a is a regular point {oTf{x) ; and
. l[d? \ f{x)ff{x){x-af \-[
when a is a pole of /(as).
68] THE EXPANSION OF FUNCTIONS IN INFINITE SERIES. 105
Example 1. Shew that
when — 1 < ^ < 1.
^"sVr+^j'^iTid+s) "^ 27176 (i^-^^ "*■•••'
Shew that the seoond member represents - , when | a? | > 1.
Example 2. If S^^ denote the sum of all combinations of the numbers
2«, 4«, 6«,...(2n-2)«,
taken m together, shew that
« sin«^^(2n-|-2)! l2n-|-3 2n-|-l ^-^ 3 p®"^^^ '
the expansion being valid for all values of z represented by points within the oval whose
equation is | sin f | « 1 and which contains the point ;?=0. (Teixeira.)
68. Expansion of a function of a root of an equation, in terms of a
parameter occurring in the equation.
Now consider the equation
0{x)^(x-a)0i{x) = t,
where ^ is a number such that along the contour 8 we have |^ (^)| > |^|> and
along the contour s we have \0(z)\ < \t\.
The equation (x) » t, regarded as an equation in x, will then have a
single root in the ring-shaped region bounded by the curves S and s; we see,
in £Btct, from the equations
2m}a0W^t-2mlJsW) Js^) ^'"i
-1,
-0,
that the equation in question has one root in the interior of S and none in
the interior of s.
Then if the function /(x) is regular in the region limited by 8 and «, we
see from the preceding articles that the formula
where An and Bn have the values already found, gives the expansion in
powers of t of the function /(a?) of the root considered.
As an example of this formula consider the equation {x - a) oosec x^^t, and let
f(-r)=-i-.
^ ' x-a
106 THE PROCESSES OF ANALYSIS. [CHAP. VL
Then we find
. cosa
" Sin a'
*" (n+l)ln ^-+1 '
* sin a' * '
Hence
cosa * t^ cP*+^(8in*a) . 1
x-a sin a n-i(w + l)In cto»*i <sina'
and thus gives the expansion, in ascending powers of t, of , where x is given in terms
of t by the equation
0? = a + < sin ^. (Teixeira. )
69. Lagrange's theorem.
Suppose now that the function /(^) is regular at all points in the interior
of £•, 80 that the poles 6n &s, ... &a do not exist. Then the formulae which
give the quantities An and 5« now become
1 d*-»
m <"-'•
^0 =/(a),
Moreover the contour s can now be dispensed with, and the theorem of
the last article takes the following form :
Let /(z) be a regular function of z at all points in the interior of a
contour fif, and let 6 (z) be a regular function with no zero in the interior of S.
Let a be a point inside iS, and t a number such that for all points z on S wo
have
\(z-a)e(z)\>\t\.
Then the equation (z — a)0 (z) = t will have one root x in the interior of
S, and /(a;) will be given as a power-series in t by the expansion
/(x)^f(a) + l -,J^
«— 1
r.
e^(ax
This result was published by Lagrange in 1768 ; it is usually stated in a
slightly different form, to obtain which we shall write
the result may now be enunciated as follows :
^//(^) ^^ ^ W ^ regular functions of z within a contour S surrounding
a point a, and if the a quantity such thai the inequality
t<l>{z)\<\z-a
69] THE EXPANSION OF FUNCTIONS IN INFINITE SERIES. 107
18 satisfied at all points z on the perimeter of £>, then the equation
z=sa + t<l>(z\
regarded as an equation in z, has one root in tlie interior of S: and if this root
be denoted by a?, then any regular function of x can be expanded as a power-
series in t by the formula
This result is of course a particular case of the more general theorem
given in § 68.
Example 1. Within the contour surrounding ;e—a and defined hj the inequality
|;?(«-a)|>|a|,
the equation
«-a— -=0
has one root z, the expansion of which is given by Lagrange's theorem in the form
Now jfrom the ordinary theory of quadratic equations, we know that the equation
z—a — =0
z
has two roots, namely
and our expansion represents the former of these ordy — an example of the need for care in
the discussion of these series. If however we regard the expansion as a power-series in a,
and derive other power-series from it by continuation in the a-plane, we shall ultimately
arrive at the series
« (-l)»(27i~l)! g'*
»Ii n\ (n-l)I a*»-i'
which represents the other branch of the function z.
Example 2. If y be that one of the roots of the equation
y^\-\-zf
which reduces to unity when z is zero, shew that
»(»+6)(w+6)(»+7) ^ w(w+6)(»+7)(»+8)(»+9) ^ ^
A. \ n \
41 ■ 6!
so long as \z\<^.
Example 3. If ^ be that one of the roots of the equation
108 THE PROCESSES OF ANALYSIS. [CHAP. VI.
which reduces to unity when y is zero, shew that
the expansion being valid so long as
\y\ < |(a-l)«-ia-«|. (McClintock.)
70. ItoucM*s extension of Lagrange^ a iheorem.
Consider now two functions /(^) and {z), which are regular at all points
within a contour 0, on the perimeter of which the inequality j j;.\^
A")
satisfied.
< 1 is
Then we shall shew that if the equation f(z)=:0 have p roots a^y a^, ... Op
in the region contained by C, the equation f{z) — a<f> (z) = tuUl have p roots
Oi', Oa', .•• ^'i *w ^f^ region; and for every function F(z) regular in the region
we shall have
r=l r=l n=l ^' r«l aO^ I -^^ (0.^)] )
f(z)
where ylr (z) = ^ ' .
We may note that this theorem reduces to that of Lagrange when
f(z) ^z^a and p = 1.
The result stated may be obtained in the following way :
We have 2 J'(a,')-K^. f ^'/-^M^^^^dz
{/(^)}'{/(^)-«<A(^)}J
t/wl /(^) 1/(4 / (^) - «<^ w J '
When n is large, the last integral tends to zero: we thus have on the
right-hand side a power-aeries in a, in which the coefficient of a* is
or
4 i r j^' \ F' (z) {<!> (z)]' jz - a,)"> [l
r> ^ ■ ■ ■ ; I .'' J. t w m _^„ ^s: , ^j.i., . 1 j e^i^^mf^^m
70—72] THE EXPANSION OF FUNCTIONS IN INFINITE SERIES. 10&
or I i JUl \ r (Or) 4> (ar)n
which establishes the theorem. Putting F{z) = 1, it is seen that the number
of roots a' is p.
71. Teixeira has published the following generalisation of Lagrange's theorem, the
proof of which may be left to the student. Let
where ^ (z)^ ..,<t>k (z) are regular functions of z in the interior of a contour K, and ^ is a
point inside K, Let a be a positive quantity, so small that the condition
z-t
z-t
+ ...+
a*0fc {z)
z-t
<1
is satisfied along the contour K, Then to every value of x which satisfies the condition
I J? I < a there corresponds a unique value of « in the interior of K ; and / («), where / is a
regular function at all points in the interior of JT, can be expanded in ascending powers of
X by the formula
where the summation is extended over all positive integral solutions of the equation
a + 2^+3y+...+i{-X«n,
and where
Another form of this result is
/W-/W +22 7-^ ^, {/' W *r-,. M (0},
M-K> v-0 syT*-) ' dt
where the quantities <^v,^ are obtained from the equations
72. Laplace's extension of Lagrange's theorem,
Lagrange's result can easily be extended to a case in which the given
equation is of a somewhat more general type.
Suppose that the equation
is given, and that it is desired to expand some function f{z) of a root of this
equation in ascending powers of t
If we write a + ^0 {z) = u,
the equation reduces to i^ = a + <<^ {i/r (t^)}.
The problem of expanding /(^) is therefore equivalent to that of expand-
ing/{'^(w)}, where u is given by the last equation; and this can be done by
Lagrange's theorem.
110 THE PROCESSES OF ANALYSIS. [CUAP. YI.
73, A further generoMsaiion of TayMs theorem.
The series of Laurent, Darbouz, Burmann, etc. may be regarded as extensions in
different directions of the fundamental series of Taylor. A generalisation of Taylor's
theorem of a somewhat different character to these, is furnished by the foUowing result,
the proof of which may be left to the student.
Iff{z) and B {£) are regvlar functione oft in the neighbourhood of the point zt^x, and if
6, ('^)^jj (t) dt, 3, W=//i W dt,
and generally
3n(^)=j\^iWdt,
then, for values of tin the neighbourhood of the point x^ f{t) can be expanded in a series of
the form
/(«)«=ao^W+MiW+«2^2(«)+.-+«»^« («)+...>
where
and generally
the number of differentiations in the last expression being n.
It is clear that Taylor's series is obtained from this expansion by putting B {t)=h
Example 1. Shew that
Example 2. Shew that
(Laurent, Joum. Math. Sp4c., 1897.)
Example 3. By writing B {t)^e^, obtain the expansion of an arbitrary function of f in
a series of the form
where o^^, o^ are independent of t.
Example 4. In the general result, shew that when ;r=0 we have
where
/W-2~5^ and d(«)-2^«*.
^i^and d(^)-2^,
ni ^ ' n\
(Quichard, Annates de Vic. Norm,, 1887.)
78, 74]
THE EXPANSION OP FUNCTIONS IN INFINITE SERIES.
Ill
74. The expansion of a function in rational functions.
Consider now a function f(z), whose only singularities in the finite part
of the plane are simple poles Oi, a,, Os, ...: let C], c^, ... be the residues at
these poles, and let (7 be a circle of very large radius R not passing through
any poles, so that f{z) is finite at all points in the circumference of C. (The
function cosec^ may be cited as an example of the class of functions con-
sidered.) Suppose further that at all points on the circumference of (7, the
modulus of f(z) is less than M, where if is a quantity which remains finite
when large values of R are taken.
Then jr—, I J-^ dz^snm of residues of "^^ at points in the interior
2w% JoZ — ic z -X ^
of (7
=/(a?) + 2-^^, .
where the summation extends over all poles in the interior of C.
But ±( f<^dz = —(^-^^ + —.l -^^^dz
iiriJce — m iirijo " 2in ] a z {z — w)
-^ ^ ^ ^an 2inJoZ{Z'-x)
if we suppose the function/(^) to be regular at the origin.
Now R being supposed large, I -j^^—k is of the order -^ of small quantities,
J gZyZ ^^X) JX
and so tends to zero as R tends to infinity.
Therefore on making R infinitely great, we have
= /(.)-/(0) + 2c„(^-l),
or
m.m^%^U^*i\.
which is an expansion off{x) in rational functions of ^.
If instead of thd condition |/(«)| < J^ we have the condition \f{z) \ < MB^, where J^is
finite for all values of R and n is a positive int^er, then we should have to ezpemd
F{z)dz
I
o z—x
by writing
cosec;?:
and should obtain a similar*but somewhat more complicated expansion.
Example I. Prove that
z ^ * \z-7iir nirj
the summation extending to all positive and negative values of n.
To obtain this result, let cosec z — =/(2)* The singularities of this function are at the
z
points issn^, where n is any positive or negative integer.
■■■■JHW jjiwii j^.^^Mn^^PHPK^mnm
^
112 THE PROCESSES OF ANALYSIS. [CHAP. VI.
For points near one of these singularities, put ^smr + (1 Then
/W=cosec(wir+f) l-^ilil^ ^ J- (l^-LY^
^ h positive powers of (.
z — nir nir
The residue of /(«) at the singularity nir is therefore ( - 1)*. Applying now the general
theorem
/(.)=/(0)+.c,[^ + I].
where c^^ is the residue at the singularity a^, we have
/«=/(o)+s(-i)-j^-L_+i^}.
But
/(O) «» Lt,.o - + (positive powers of «) - - =0.
Therefore
cosecfs
which is the required result.
Example 2. If a is real and positive and less than unity, shew that
6^ 1 « 2« cos 2nair - 4nir sin 2nan'
For ^f{z)^- — r — , the singularities of f(z) are at the points zs^^nni, where
n«=±l, ±2, ±3, ... ±00.
For points z near z « 2wirt, put z = 2nirt + f . Then
= — — +a series of positive powers of f.
The residue at z^s^nin is therefore e*^'*.
Also I •
/(0) = fl±5±::.-n
= [i(l+«.+ ...)(l+|+...)"-i2_^
Applying the general theorem
/W=/(0)+2c,(^ + l).
we have therefore
H±«) ^i»a 00 sin2natr
2 — T-^+ 2
EXS.] THE EXPANSION OF FUNCmONS IN INFINITE SERIES. 113
But
- 2^. log ( - «-»^) = ^. (irt - 2a,r0
1
Thus
«• — 1 z N.±i # — 27ii7r " «-i \« - 2mV z + 2niir)
"* 2^ COB 2nafl' — 4n9r sin ^nair
Example 3. Prove that
1 ^_1_ _ 1_ I _2 1
7r«*(fl«-2co8^+«~*)"2fra?* «*-«-* ir* + Jar* «*»-«-«» (2ir)*+i^
For the general term of the series on the right is
{-\Yr _ 1__
which is the residue at either of the four singularities r, -ryvi^ -ri^ot the function
vz
(ir*a^ - JjF*) {eF* - e ~ »*) sin irz '
The singularities of this latter function which are not of the type r, -r, f% —riy
are at the points
^ ±s/i X ± sT-^i X
V2 "• \/2 «■
2
At «a=0 the residue is -^;
at either of the four points «= ~ J" — , the residue is
v2 'f'
ir«~l
/Vife _>/jr
^ ) . VtJr
S/ sin ._
V2
Therefore
__ 1 /" ffgcfe
~ 2wt j c ('T*'?* - J^) (e^* - e - »«) sin irz *
where C is an infinite contour. But at points on C, this int^prand is infinitely small
compared with - ; the integral round C is therefore zero.
W. A. 8
114 THE PROCESSES OF ANALYSIS. [CHAP. VI.
1 - (-lyr 1 :ii
-1
,^(;--)^_«'-"l}{.'-'>l _,»-«!}
-1
1
ira^ (e« - 2 cos ^+«~«) '
which is the required result.
Example 4. Prove that
Example 5. Prove that
coseoh*=l - Ste (^ - ^;^ + gj^ ...) .
Example 6. Prove that
8ech;r=4ir(^^,^^-^^^ + 2j^a^^...j.
Example 7. Prove that
coth *-l + to (^ + ^^-, + g^+ . ..) .
Example 8. Prove that
2 2 7—5- — o> , g . ,gv = -7 coth ira coth irb.
(Cambridge Mathematical Tripos, Part I, 1899.)
76. Expansion of a function in an infinite product
The theorem of the last article can be applied to the expansion of
functions as infinite products.
For let / (z) be a function, which has simple zeros at the points Oi , a,, 0,, . . .
where Limit | On | is infinite ; and suppose that f{z) has no singularities in
the finite part of the plane.
Then clearly /' (z) can have no singularities in the finite part of the
f (z)
plane, and 80*^-77^ can have singularities only at the places Oi, a,, a,, ....
75]
THE EXPANSION OF FUNCTIONS IN INFINITE SERIES.
115
Now for values o(z near Or, we have by Taylor's theorem
/(z) = (^ - ar)f (a,) H- (l^f' (a,) + . . .
and /' (^) =/' (a,) + (z^ Or)/" («r) + • • • •
Thus we have
:r7\ ~ ^ * constant + positive powers of (z — Or).
J yZ) Z — Or
f'(z)
At each of the points Or, the function ''-^it^ has therefore a simple pole, with
the residue + 1.
f (z) . .
If then -zj-r has at infinity the character of the functions considered in
the last theorem, it can be expanded in the form
Integrating this expression, and raising it to the exponential, we have
where c is a constant independent of z.
Putting 5 ~ 0, we see that/(0) = c, and thus the general result becomes
f'io)
This furnishes the expansion, in the form of an infinite product, of any
function /(z) which fulfils the conditions stated.
This theorem is a case of a general theorem on the factorisation of functions, which
is due to Weierstrass, and which will be found in Forsjrth's Theory of Functions,
Chapter v.
Example 1. Consider the function /(«)<
z
, which has simple zeros at the points
rn-, where r is any positive or negative integer.
In this case we have /(O) =« 1 , /' (0) = 0,
and so the theorem gives immediately
sin
^'-.?«{(-4)-l'
since the condition relative to the behaviour of "^tt-t at infinity is easily seen to be
fulfilled.
8—2
116 THE PROCESSES OF ANALTSia [CUAP. VI.
Example 2. Prove that
h©} {-(s.!.)} {-(s^.)} {-U^j} (-(^^1
cosh k - COB X
1 — oosx
(Trinity College ExaminatioD, 1899.)
76. Expansion of a periodic function in a series of cotangents.
Another mode of expansion, which may be applied to periodic functions
whose poles are all simple, is that indicated in the following example.
Consider the function
cot {x — Oi) cot (a: — a,) ... cot (x — On).
This is a trigonometric function of a?, having poles at the points Oi, a,, ... On,
and also at all other points whose afiSxes differ from one of these quantities
by a multiple of tt. There is clearly no loss of generality in supposing that
the real part of each of the quantities Oi, a,, ... 0^, lies between and tt.
Now let ABCD be a rectangle in the j^-plane whose comers are the points
J[(^ = — too), £(^sB7r— 100 ), C(j?=7r-»- too), and D(^ = too); and consider
the integral
^r — A cot (if — Oi) cot (if — Oj) ... cot (-^ — On) cot (j? — a?) ck
taken round the perimeter of the rectangle.
The integrals along DA and CB are equal but of opposite sign and cancel
each other. Along CD, each of the cotangents has the value - 1, so the
integral along CD is . Similarly the integral along AB has the value
•5- . The whole integral has therefore the value
l+(-l)\.,
%\
2
The singularities of the integrand in the interior of the contour are at the
points 2^ = Oi, Os, ... On, X ; and clearly the residue at tir is
cot(ar — cti)cot(ar — aa)...cot(a,. — ar-i)cot(ar — Or+i) •••
cot (ttr — On) cot (Oy — x),
while the residue at a; is
cot(a: — Oi) ... cot(a? — ttn).
Since the value of the integral is equal to the sum of all these residues, we
thus have
1 4.(- l)n • r=i»
^— ^ t** = cot(a? — cti) ... cot (a? — On) + 2 cot(a,. — Oi) ...
cot {Obr — a>f^ cot (Or — x).
76, 77] THE EXPANSION OF FUNCTIONS IN INFINITE SERIES. 117
Thus if n be even, we have
cot (a: — Oi) ... cot(a? — On) = 2 cot(ar — ch) • ••cot (or — On) cot (a? — Ctr) + (—l)*,
and if n be odd we have
cot(d? — Oi) ... cot(a?-a„)= 2 cot (ctr — cti) . . . cot (cv — On) cot (a? — a^).
This method of decomposition into a series of cotangents is of very
general application to periodic functions ; it may be regarded as the trigouo-
metrical analogue of the decomposition of a rational function into partial
fractions.
Example. Prove that
8m( 3? - ^i) sin (a? - fcj) • • • ^^^ (^ ~ K) ^^ (<*i "" ^i) • • • ^^^ (*i " ^1*)
Bin (47 -a^) sin (a? -Oj) ...8in(a:-a„) sin (aj-Oj)... sin (aj-oj
cot(4;— Oj)
8m(aj-aj)...S]n(a2-aJ ^
+
+ C06 (a|+a2+...+a„-fci — 6j— ... — 6i»).
77. Expansion in invei^se factorials.
Another mode of development of functions, which although investigated
by Schlomilch as long ago as 1863 has hitherto not been much used*, is that
of expansion in inverse factorials.
Let Z be a line drawn parallel to the imaginary axis in the j^-plane ; and
draw a circle of large radius, having its centre at the point where I cuts the
real axis.
Consider a function f{z\ which has no singularities within the semi-
circular area which is bounded by I and this circle and which lies on the
positive side of Z ; let 7 be the semi-circular arc which bounds this region.
Suppose moreover that at all points of 7 we have the inequality
\f{z) i < M
satisfied, where M is finite however large the radius of 7 may be chosen.
Then if z be a point within this semi-circular region, we have
Now
[ f(t)dt ^r f(t)dt r zf(t) dt
♦ Beferenoes to some recent work are given by Klajver, CampUs Rendui, oxxxir. (1902), p. 687.
^^1
— - ^> K*
118
THE PROCESSES OF ANALYSIS.
[chap. VI.
But
/,
zf{t)dt
yt\t-Z)
< \z
m\
dt
y\t\\t-z\'
which is infinitesimal when the radius of 7 is infinitely great.
Thus
if we now suppose that the direction of integration along I is from — loo to
+ 100.
Now if n be any positive integer and z be not equal to 0, — 1, — 2, etc.,
we have the identity
1 1
+ ...+
z-t -^ • -8^(^+1) ' ^(^-i-l)(^ + 2) z{z-k-l)...{z + n){Z'-ty
on substituting this in the second integral we have therefore
Oi
y(-^) = ao + ^ +
(h
z ziz-^-l)
+ ...+
^'Wi
where
z{z-k-\) ,,.{z -^-n)
■^ 2in}iz (z + 1) ... IzJf n) (z - 1) '
/(t)dt
t
^=2-^//(^>^^'
Now the product
can be written
t(t+l)...(t + n )
z{z-{'l)...(z + n)
t **
-n
r
and it diverges to zero or to infinity when z tends to 00 according as the real
part of t—z is negative or positive, as can be seen by comparing it with
the product
77] THE EXPANSION OF FUNCTIONS IN INFINITE SERIES. 119
which has the value (n+ iy~'. But the real part of ^ — <« is, in the case
under consideration, negative ; and so the product
t(t+l)...(t + n)
is infinitesimal when n is infinite.
Since/(^) is finite along I, and / . *^' . is finite, we see that
I z{Z'\-l)...(Z'^ n){z^t)
is infinitesimal when n is infinite.
We can therefore expand f(z) in the form
>^<^>="«+ 7 + ^(^Ti) + ^(7+lHl+ 2) + - •
the coefficients a being given by the above equation ; and this expansion is
valid for all values of z whose real part is greater than the real part of z at
any of the singular points of f(z), except for the points
^ = 0, -1. -2
Example 1. Obtain the same result by using the equalities
«(«+l)(f+2)...(«+n) n\Jo ^ «/-««, ,
Example 2. Obtain the expansion
where a„= r<(l-<)(2-<) — (^-l-0«fe>
and discuss the region of its convergency. (Schl6milch.)
Miscellaneous Examples.
1. Let er^Pn denote the nth derivate of e'^y so that
P^,= l, P,= -2«, P,-4a»-2, etc
Shew that if f(z) is an arbitrary function, then/(«) can be expanded in the form
1 r*
where a-= | e-^Pn(x)f(x)dx.
2.4.6... 2nVir; —
and find the region of convergence of this series. (Hermite.)
120 THE PROCESSES OF ANALYSIS. [CHAP. VI.
2. Obtain (from Darbouz's formula or otherwise) the expansion
+ ;
find the remainder after n terms, and discuss the convergence of the series.
3. Shew that
+( - D— ^•^•°-;-,j^""^^ J {/" (*+ A)+( - !)«/• (*)}
( - l)»h<'*' j\,{t)f*'{x+/U) dt,
where y, (*)= -^^i-^ x»+i (\-x)n+i^{x-i (1 -a:)-*}
rrfilj
and shew that y^ (x) is the coefficient of n ! ^ in the expansion of {(1 — tx) (1 +<— tir)}'i in
ascending powers of t.
4. By taking
in Darboux's formula, shew that
/(*+A) -/(x)= -a,h {/' (*+A)-l/' (*)|
V 1 — r w tt' u'
MISC. EXS.] THE EXPANSION OF FUNCTIONS IN INFINITE SERIES. 121
5. Shew that
2!
~ ^^»^^;P<'-° ^r'(a)4-/-(.)}
+
271 !
JV2m(0/<*'**U«+<(^-«)}<'^
-^ ♦.<')%-|r[^^'.(.t;)]...-
6. Prove that
+ (7«(«,-^)V'M^)+
where CJ^ is the coefficient of 2!* in the expansion of cot (^-5)1^ ascending powers of z,
(Trinity College Examination.)
7. If ^1 and x^ are integers, and <^ (2) is a function which is regular for all values of z
(finite or infinite) of which the real part lies between x^ and x^, shew (by integrating
/
4>{z)dz
£>2iriz _ 1
round a rectangle whose sides are parallel to the real and imaginary axes) that
i*(*'i)+*(^i + l)+<^(a?i + 2)+...+<^(a?,-l)+i<^(^^
= ['*<f>{z)dz'k-l /'" ^(^tH-<y)-»(:Ci4-*y)-»(^,-ty)+»(j?,-ty) ^^
Henoe by applying the theorem
where i?i, ^3, ... are the Bemoullian nimibers, shew that
^(l)+<^(2)+...+<^(n)-C+i^(n)+|**(«)c?^+^|^^-^^^^^
(where (7 is a constant not involving n) provided that the last series converges.
8. Obtain the expansion
for one root of the equation x^2u+u\ and shew that it convei*ges so long as | :p | < 1.
122 THE PROCESSBS OF ANALYSIS. [CHAP. VI.
9. If /S^i, denote the sum of all combinations of the numbers
1«, 3«, 6«, ... (2n-l)«
taken m together, shew that
* ~8in»^^(2n+2)! l2«+3 "ifn+i) 2»+l* •" ^ »(»+« Sj
10. If the function /(<) is r^[ular in the interior of that one of the ovals whoee
equation is | sin2|>'C (where C^ 1), which includes the origin, shew that /(<) can, for all
points < within this oval, be expanded in the form
- /P»)(0)+;S*2/(*-»(0)+...+^"-»/<'>(0)
n')-m^^r — 2^1 '^•"'
. g /'--''(0)+^^,./O'-')(0)+...+C/'(0) ^^....,
where ^S^' is the sum of all combinations of the numbers
2«, 4«, 6«, ... (2n-2)*
taken m together, and £f^^^^ denotes the sum of all combinations of the numbers
1«, 3«, 6«, ... (2n-l)«,
taken m together.
11. Shew that the two series
2f^ 2^^
d 2g 2 / 2g Y 2 . 4 / 2^ y
represent the same function in one part of the plane, and can be transformed into each
other by Burmann's theorem.
12. If a function f(z) is periodic, of period 2fr, and is regular in the infinite strip of
the plane, included between the two branches of the curve | sin «|kC7 (where 6'>1), shew
that at all points in the strip it can be expanded in an infinite series of the form
/($)= Aq-^- Aisin z-^ .,, + A^ain^ z-^
+coe «(i9i+5j sin «+...+5i»8in«-i «+...);
and find the coefficients A and B.
13. If <f> and / be connected by the equation
of which one root is a, shew that
^W ^ 1 ^^'•'^ ^2! <^'» <1>"{PF')'
1
1
^ JL
31 <^'«
4>"' (*T' ( /'^T
1.1.2
T .••*.•,
MISC. EXS.] THE EXPANSION OF FUNC?riON8 IN INFINITE SERIES. 123
where Fy /, F\ etc. denote
F(a\ f{a\ -^-
14. If a function If (a, 6, x) be defined by the series
IF (a, 6, ^)=a:+ 0-7-^+
which converges so long as
2!
x\<
31
*•'+
h\:
shew that dx^^""^ *' ^)-l+(a-fc)ir(a-6, 6, d?);
and shew that if y « ir(a, 6, x)y
then
a;» IT (6, a, y).
Examples of this function are
ir(l, 0, ^)=^-l,
If (0, 1, a:)=log(14-a;),
(l + ;c)«-l
15. Prove that
If (a, 1, 0?)^
1 1 . ; (-ir^ g
00
n—O
where
Q.
4a^
3a,
6a,
4a|
3ao
(2n-2)aH-i : (n-l)ao
na^
(n-l)a^., a.
and obtain a similar expression for
< 2 a,idf*> .
(n-o ;
(Jeiek.)
(Mangeot.)
16. Shew that
2 CLfSf
where /S^ is the sum of the rth powers of the roots of the equation
2a,.a?*'«*0.
(QambrolL)
17. If /»(«) denote the »th derivate of /(«), and if /-»(«) denote that one of the nth
integrals of f{z) which has an n-ple zero at < *0, shew that
and obtain Taylor's series from this result, by putting g (s}b1.
(Quichard.)
124 THE PBOCESSES OF ANALYSIS. [CHAP. VI.
18. Shew that, if ^ be not an integer, the series
2
(xH-m)*(arH-n)*'
in which m and n receive in every poesible way unequal values, zero or integers lying
between +/ and -/, vanishes when /increases indefinitely.
(Cambridge Mathematical Tripos, Part I, 1895.)
19. Sum the infinite series
1
n—
where the value n=0 is omitted, and p, q are positive integers to be increased without
limit
(Cambridge Mathematical Tripos, Part I, 1896.)
20. If F(x)^ei^'"^^'^^^, shew that
n-l
and that the function thus defined satisfies the relations
F{x)F(l-x)^2amxw.
Further, if ^(r)=:«+^ + ^,+
-J^og(l-«)rf(log4
shew that /'(ar)=c*'^2^*<^~'''*^^
when I !-«-«»*« I <1.
(Trinity College Examination.)
21. Shew that
[-(r][-(i^)l-G^J][-G-^J][-GA^)"]
n
I
2«(1-C08A')*«" *
where (Lf^Jc Bin -^ — ir,
n
MISC. EXS.] THE EXPANSION OF FUNCTIONS IN INFINITE SERIES. 125
" n
and < j: < 2ir. (Mildner.)
22. If I ^|< 1 and a is not a positive int^er, shew that
where (7 is a contour in the ^plane enclosing the points 0, x, (Lerch.)
2a If ^(«), <^,(«), ... are any polynomials in z, and if F(s) be any function, and if
^1 (')) ^a (^)) ••• ^ polynomials defined by the equations
[* ^(4r) ,^, (*) *, (X) ... <^,_, (*) *"> )-»» W d«=^„(,),
y a a— ar
shewthat P ^W^ ,^iW, ^^aW
*i W 4*% («) *s W
1
rF(x)<h{x)<h(x).,.4,^{x)^.
J a t—X
24. A system of functions p^ (z\ p^ (z\ p^ (z), ... is defined by the equations
where a» and b^^ are given functions of n, which for n^oo tend respectively to the limits
Oand -1.
Shew that the region of convergence of a series
where e^y e^y ... are independent of «, is a Cassini's oval with the foci +!> - 1.
Shew that every analytic function f(z\ which is regular in the interior of the oval, can
for points in this region be expanded in a series
where
126 THE PROCESSES OF ANALYSIS. [CHAP. VI.
the integrals being taken round the boundary of the region, and the functions q^ {z) being
defined by
26. If Pn (*) be the coefficient of — j in the expansion of
in ascending powers of z, so that
shew that
(1) Pn {x) is a homogeneous polynomial of degree n inx and A,
(2) 2-^-- (^^^)'
(3)
rPn{s)dx^O (n^l),
(4) If y — o^Po (x)-^aiPi (^) H-OjPj (*) + .. ., where a^, o^, a,, ... are real constants,
then the mean value of -j^ in the interval from x^ - A to x^m 4.A is o^. (L^ut^)
26. If P« (j?) be defined as in the preceding example, shew that
/>^-(-l)-2^^(^cos-j-2,^cos-^+3^cos-^ +...J,
i'2»+i = (-ir2^^:^j(^sm-^-2^^jSm-^ + ^5;^jSm-^ + ...j. (AppelL)
i
CHAPTER Vll.
Fourier Series.
78. Definition of Fourier series; nature of the region within which a
Fourier series converges.
Series of the type
Oo + a, cos ^H-ag cos 2-^ + 08 cos 3£r + ... + 6i8iiiz + 6,sin 2^ + 63sm 3£r + ... ,
where Oo, Oi, a,, a,, fej, ftj, 6,, ... are independeat of z, are of great import-
ance in many analytical investigations. They are called Fourier Series,
We have already seen that the region within which a series of ascending
powers of z converges is always a circle \ and the region within which a
series of ascending and descending powers of z converges is the ring-shaped
space between two circles \ we are therefore led by analogy to expect that
series of the Fourier type will likewise converge within a region of some
definite character.
To investigate this question, write 6**= f.
The series becomes
This is a Laurent series in f ; it will therefore be convergent, if at all,
within a ring-shaped space bounded by two circles in the ^-plane ; that is,
it will be convergent for values of f satisfying an inequality of the type
•
a<|?|<fe,
where a and h are positive constants.
Now let
£r = a: -h ty ;
then
128 THE PROCESSES OF ANALYSIS. [CHAP. VU.
aod therefore the inequality becomes
log a< — y< log 6.
This inequality defines a belt of the ^r-plane, bounded by the two lines
y = — log a and y = — log h ; hence the region of convergence of a Foui-ier
series is a belt of the z-plane, bounded by two lines parallel to the real axis.
It may however happen that the Laurent series in f is divergent for all
values of f, in which case the Fourier series is divergent for all values of z ;
or, (and this is the most important case for our purpose,) it may happen that
a » 6, so that the region of convergence of the Laurent series narrows down
to the circumference of a single circle in the ^-plane ; in this case the region
of convergence of the Fourier series narrows down to a single line parallel to
the real axis in the plane of the variable z.
If now the coefficients Oq, Oi, a,, ... 6], &t, ... are all real, considerations of
symmetry shew that if the Fourier series is divergent for a value z^a + tb,
it will also be divergent for the value -zr =s a — i6 ; so if in this case the region
of convergence narrows down to a line, that line can only be the real axis in
the ^-plane.
Hence a Fourier series with real coefficients may converge only for real
values of z, and diverge for all complex valves of z.
An example of this class of expansions is afforded by the series
sin-^ — 2®^^2^+3^^3^"" 4 s^^ 4^+ ... .
Writing this in the form
we see that it diverges when z is not purely real ; when z is purely real and
not an odd multiple of ir, the sum of the series is
ilog(l+6'0-^.log(H-6-^X
or 21'^^^**'
or i^z-Vkir,
where k is some integer, as yet undetermined.
Now when z = the sum of the series is seen directly to be 0; when
z^-ci. the sum of the series is tan~^ 1, or 7- ; when ^ = — — the sum is
2 4 2
78] FOURIBR SERIES. 129
— tan""* 1, or — 7 . In this way we see that when z lies between — ir and + ir,
the integer k is zero.
But A? is no longer zero when z is greater than ir ; for each term of the
series is clearly unaffected if ^ + 27r be written for z : hence the sum of the
series must be the same for -^ + 27r as for ir ; and hence when wkzk Stt,
the sum of the series is ^z-^tt; so that when z lies between ir and Stt, the
integer Ar w — 1.
Proceeding in this way, we see that the sum of the Fourier series is
^z + kir, where k is an integer chosen so as to make ^z + kir lie between
— ^ and + ^ . This is important as shewing that the sum of a Fourier series
is not necessarily a continuous analytic function. It is clear however that the
sum of a Fourier series can have discontinuities only in the case in which the
region of convergence narrows down to the real axis ; in the other case when
the region of convergence is a belt of finite and infinite breadth, the Laurent
series in ^ represents an analytic function, and therefore the Fourier series in
z does also.
Example. Shew that the series
cos «- 52 cos 2« + 5j COS3-5- ...
n^ 1
converges only for real values of 2, and that when -ir<«<-firit8 sum i^ ts - 7 **•
For when z is real, the series is absolutely and uDiformly convergent, as is seen by com-
paring it with the series 1 + 02 + 02 "*"••• •
When z \a complex, we have (putting z=x+iy)
-ijC0srw=^ {e*(»«+"*»')-|-e<(-»«-»«*')} ;
now either — , or ^ s ^ infinite for n=ac, so the terms of the series are ultimately
infinitely great and the series diverges.
To find the sum when z is real, it has been shewn that when -7r<z<ir we have
^z=:8in z- ^ sin 2i+i sin 3;? ... .
This series is imiformly convergent in the interval (though not at its extremes - ir and fr)
and so can be integrated.
Thus c-J«*=cos;8-5jCos2a+^cos3«-,..,
where c is a constant.
W. A. 9
130 THE PROCESSES OF ANALYSIS. [CHAP. VII.
To find c put «»0, which gives
whence the result.
,1.1 _^
^" 2«"*"3«"*""12'
79. Valties of the coefficients in terms of the sum of a Fourier series, when
the series converges at all points in a belt of finite breadth in the z-plane.
The connexion between the coefficients a^, Oi, a,, ... , 6i, 69, ... of a Fourier
series, and the sum of the series, can be easily found in the case in which
the series converges in a belt of finite breadth in the ^-plane. For in this
case, as we have seen, the sum of the series is an analytic function of z. Let
it be denoted hyf{e), so that
f(t) ss a^ + a^cos z + a^coa 2z + ... +6, sin 5-^69 sin a? + —
Writing f = e^, the series becomes
and by Laurent's theorem the coefficients in this expansion are given by the
equations
2^^-^^j /(^)?-»d?.
where C is any circle in the (-plane, surrounding the origin and contained
within the ring-shaped region in which the expanded function is regular.
Now if the quantities Or and br are all real, we see as ^before by symmetry
that the real axis must be contained in the region of convergence in the jr-plane,
and therefore the circle of radius imity must be contained in the region of
convergence in the (-plane, since this circle corresponds to the real axis in the
jff-plane. We can therefore take (7 to be a circle of radius unity, with the
point (b as centre.
Now writing f « e^ in the integrals, we have
and so
(ar + ibr)^j^y(z)(f^dz,
1 r*»
ttr = - / f(z) COS rzdz (r > 0),
1 ri» .
6r « — I f{z) sin rzdZf
79, 80] FOURIER SERIES. 131
and ^"""^ZttJ f^^^^'
These equations give the vahies of the coeflScients Oo, Oj, a,, ... ,
bi, ftj, ... , of the Fourier series, in terms of the sum f(z) of the series,
in the case in which the series converges over a belt of finite breadth in the
2:-plane. We shall see in the next article that the same formulae hold good
in the more extended case, in which the series converges only for real values
of z.
sme
JBaxunpU, Shew that the function , f^ can, when k<l, be expanded in a
Fourier seriee of sinee of multiples of «, valid for all points z situated in a belt, of width
-2 logir, parallel to the real axis in the ;B-plane.
For we have
sin a
= Li_i 1 .1
00 I
and this can be expanded in the form 2 i, /;* (6***-e-**»), provided k le**! and ;t |e-*»| are
less than unity. This can only happen when their product X^ is less than 1, Le. when
-l<it<l.
When this condition is satisfied, on putting z^x-^iy, it is clear that we must have
e*«-i'|<T and>it, Le. we must have-y lying between logfi) and log it, Le. z must be
within a belt of width -21ogiI;, parallel to the real. axis. When these conditions are
satisfied the expansion is valid, and so
8in« * . , .
2 A!* *smn«.
l-2irco8;j+it*
Hal
80. Fourier's Theorem.
ft
We have already said that the most interesting cases of Fourier's series
are those to which the investigation of the last article cannot be applied,
on account of the fact that the series converges only for real values of z. It
is therefore necessary to undertake another investigation, in which the
assumptions of the last article are no longer made. The result to which we
shall be led is known as Fourier's theorem, and may be stated thus :
Iff (z) be a quantity which depends on a variahle z, and which is finite and
has only a limited number o/maaima and n^inima and of finite discontinuities
in the interval <z< 2ir, then the sum of the series
CO
^0+ S (amOosmz + bmsiamz),
mai
9—2
132 THE PROOBSSES OF ANALTSia [CHAP. VII.
where
1 f *»
dm = - / /(t) COS mtdt,
1 r*»
6m= - / /(O sin TYVtdt,
repre^ento y (2:), at every point in the interval < ^ < 2^ for which f(z) is
continuous; and at every point in the interval 0<z<27r for which f{z) is
discontinuov^Sy the sum of the series is the arithmetic mean of the two values of
f(z) at the discontinuity.
The diacussioQ of Fourier's theorem given below is a modification of what is known as
Cauch^i second proofs which was originally published in 1827 in the second volume of his
Exercices de Math^moOiqueif and is reprinted in his Collected Works, Second Series, YoL vn.,
p. 393).
This proof, (which in its original form was in some respects imperfect,) seems to have
been little used by the mathematicians of the nineteenth centiuy, who in the discussion
of FourieHs theorem almost universally followed the exposition of Dirichlet (which is
also reproduced later in this chapter) ; the importance of Cauchy's proof was shewn by
A. Hamack in 1888. It may be observed that the restrictions placed on /(«) — as to its
having only a limited number of maxima and minima, etc. — are suficieni but not necessary
for the validity of the expansion.
To establish the theorem, we write the first 2ik + 1 terms of the expansion
in the form
~l f(t)dt'^^ 2 f{t)coam(Z''t)dt,
or
or . Uk-h Fjk,
where Um = "*2* ^ f V* <*- V(0 dt,
We shall now investigate the behaviour of the quantity J7fc when k, though
finite, is a large number.
Let <t> (^) denote the quantity
80]
FOURIER SERIES.
133
Then <f> (^ clearly has a definite value corresponding to every value of ^,
except the exceptional values ^=0, ±i, ± 2t, for which c*^= 1 ; moreover,
it is easily seen that the quantity
tends to a definite limit when B^ tends to zero, independently of the way in
which Sf tends to zero (still excepting the points 0, ±i, ±2i...); hence
<f>{^) is an analytic fiiDctioD of f, having poles at the points 0, ± t, ± 2i ... ;
and the series U^ is clearly the sum of the residues of ^ (^) at those of its
poles which are contained within a circle C> in the ^-plane, whose centre is
at the origin and whose radius is (* + 2) •
Hence
Write
Thus
Now we can write
.i-»-»
+»-» j^.i-i »jr+»-i
ir , , _
^*==2Uf +f +f +r +r u*(^^^
«/i + /, + /3 + /4 + /b say.
At points in the range of integration of /, and I^, the real part of ^is
positive and at least of order k'^; and so for these integrals we have
In this expression, as k tends to infinity, ^^-^^^ tends to the limit unity,
and {'e^(»-<-*r) tends to the limit zero : thus ^<f>(^ tends to the limit zero,
since the range of integration from to ^ is finite ; and hence as /i and I^
are the integrals of ^(f> ((^ taken over finite ranges, we see that /i and I^ tend
to zero as k tends to infinity.
Considering next the integrals /, and /«, we observe that the quantity
^^_^ is never infinite when < ^ < 2: < 2'jr, and so f^ ((^ is never infinite ;
134 THE PROCESSES OF ANALYSIS. [CHAP. VIL
and thus, since /{ and /« are integrals of ^(f> (^ over ranges which become
infinitesimal as k tends to infinity, it follows that /, and /« tend to zero as k
tends to infinity.
Consider next the integral /j, or
where f* (?) « ^^jjeii^t,f(t) dt
In the range of integration of /«, the real part of ? is negative and at
least of order W. The feu^tor -r-? — r- therefore tends to the value — 1 as fc
6*^— 1
tends to infinity ; denote it by — (1 + aj^), where o» tends to zero as k tends
to infinity.
Now r^(^^*^/(t) dt^J, + /„
^0
where /,«[ ^'^ l:^^'^fit)dt,
and t7,= r ^ K^^-^f{t)dL
'"log*
Considering first /i, we see that within its range of integration the
quantity ^{z-^t) has its real part always negative and at least of order
'. — T, which tends to infinity with k\ hence the quantity ^ef^^^^ tends to
zero as k tends to infinity ; and therefore as the range of integration in t/^ is
finite, we see that Ji tends to zero as k tends to infinity.
Consider next J^. Writing t; ■» ? (-«r — e), we have
log*
and writing e^^Wy this becomes
/•log* / ^\
J.^j^''f[z-'2^)d..
Now as k tends to infinity, 6*^* and — ^ tend to the limit zero. Let
/(^— 0) denote /(^) if ir is a point at which the function f{z) is a continuous
80] FOURIER SERIES. 135
function, and at those points at which f(z) is a discontinuous function let
f(s — 0) denote that one of the two values oif{z) which is continuous with
the value of/ for values smaller than z. Then since there cannot be another
discontinuity within an infinitesimal distance of z, we can write
/(-^)'/(*-0) + ,.
r
where 17 tends to zero aa k tends to infinity ; and so
f -L.
[e fe
t7,=/(^ — 0)1 dw+l lydiw;
jf - log*
= -/(^ - 0) + e'^^'fiz - 0) + j ^ vdw,
or J,=:-/(^-0) + €,
where e tends to zero as k tends to infinity.
Thus
l;<l>(0 (l + «*){/i-/(^-0) + £},
where a», Ji, and e, each tend to zero as k tends to infinity. We can write this
^<f> ((^ ^/(z — 0) + T, where t tends to zero as k tends to infinity ; and this is
true throughout the range of integration of the integral is-
Thus
or /, = 5 /(z — 0) + a, where <r tends to zero as k tends to infinity.
Hence finally
I7» = 5/(-^-0) + <r+/, + /. + /4 + A,
where <r, /i, /,, /«, I^ each tend to zero as k tends to infinity ; which can be
written
where Uj^ tends to zero as k tends to infinity.
Similarly we can shew that
where v^ tends to zero as k tends to infinity, and where /{$ + 0) denotes f(z)
136 THE PROCESSES OF ANALYSIS. [CHAP. VH.
if 2^ is a value for which the function/(^) is continuous, and denotes the value
oif{z) for values slightly greater than 2: if 5 is a value for which the function
f{z) is discontinuous. Hence the sum of the first (2A; + 1) terms of the
Fourier series is
\f{^ - 0) 4- ^/(^ + 0) + w* + V*,
where uj^ and v* tend to zero as k tends to infinity ; the sum to infinity of the
series is therefore
^{/(^-0)+/(^ + 0)].
which establishes Fourier's theorem.
It must be observed that the sum of the series coincides with f{z) only
far values of z between and 27r ; outside these limits the sum S (z) of the
series can be found from the circumstance that S{z-\'2nir) = S(zX (a result
which is obvious, since all the terms are periodic) ; while f{z) may of course
have any values whatever when z is not included between the limits and 2^,
Example, Take a function /(a) such that
IT
f{z)wm- from «««0 to «=ir,
and /(«) = -J from;? « IT to «=2ir.
The corresponding Fourier series is
Oo + SOm OOS97U + 26m siu mz,
where O'm^^- \ f(t)co3mtdt,
n J
'm
1 r«»
-i /(Osi
irjo
sin mtdt,
These integrals give
1 f» 1 f^
00=0, Otn'^i I cosmtcfe-- / cosmtdt^'Oj
6«=T / sinm^(ft-7 I nin mtdt=-r- (I -cos mw).
4;o 4j» 4m''
Therefore 6»,=0 if wi ia even, and 6„»= — if m is odd ; and so we liave
J, . sinz sindi? . sinSz .
which is the required Fourier expansion.
81] FOURIER SERIES. 137
This series can be summed by elementary methods in the following manner. We
have
. ^sinat^sin52 1 ( ^^^ ^ \ 1 / ^^«-w^ \
4i*^(l-e<')(l+fl-^) 41^"^ 4+ 2 »
where r is an undetermined integer. It is clear from the above that r actually has the
value zero when < 2 < ir, and unity when w<z<2w.
81. The representation of a function by Fourier series for ranges other
than to 2^.
Suppose now that the range of values of z, for which it is required to
represent a function f(z) by a Fourier series, is not the range from to 2^,
but from a to b, where a and b are any given real numbers. To extend
Fourier's result to this case, we take a new variable z defined by the
equation
and write
6 — a /
•^(«+-^^)'^<'^-
Then F{/) is a function whose value is given for all values of its argument
y between and 2^.
Therefore by the previous result we have
^(i^ = A fV(Od^+- 2 f*'cosm(/-0^(0^',
or writing
we have
6 — a ,
/«-^J>)*-E^,.!j> ^™t5-/«^
This last result may be regarded as the general form of Fourier's theorem.
• Example. To express the function — ^ — ^^ as a Fourier series, valid when
-ir<z<ir.
Here a™ — ir, 6«»r.
138 THE PROCESSES OF ANALYSIS. [CHAP. VII.
The formula therefore becomes
/W-i r /(0^+; 2 f' coBn{z-t)f(t)dt.
^*r y -r w nail J -»
Since in this case / (O— -/( - 0» ^^ reduces to
/(«)=- Z sin Yu / — --sinn^(i<
•■ 2 sm nj I ^ . . ^i _^ ~ dt
ii.iirt («**»- e~"^) I m+tn m-in J
• (-l)»sin nz / 1 _ 1 \
HBi »r(m'+n«) ^
which is the required expansion.
82. The Sine and Cosine Series,
We proceed to derive two particular cases of Fourier's theorem which are
of frequent occurrence.
Suppose that a function f(z) is given for a range to 2 of values of the
variable z, and that we require a series which shall represent f(z) for these
values of z^ and which shall have the value /(— z) for values of z between
and — L
To obtain a series of this character, we write in the preceding result
a » - Z, h^ly /{—z) ^f(z). Thus we have
/(*) - ^ijjit) dt + ] J J'^cos^'^-^l^- *>/ (*) dt,
/ w - i/Vw '^ + T A «» ^£ «» ^y (0 dt.
which is called the Cosine Series.
If on the other hand we require a series which shall represent /(^) for values
of z between and I, and shall have the sum — /(— z) for values of z between
and — I, we write in the general result a = — Z, 6 = hfi" z) =» — /W> and
thus obtain
f(z) « 7 2 sm -J- I sm j-f(t) at,
^ m-l f' Jo ^
which is called the Sine Series,
82] FOURIER SERIES. 139
If ^ z
EsxanpU 1. Expand -—^ sin « in a cosine series, valid when 0<f <fr.
When < f < IT, we have by the formula just obtained
-sinf-s^- I (w-t)smtdt+- 2 cosmi / {w''t)BmtooenUcU
«r y IT iM>i J
2
•i[o"^'"~']"i/o'^^^
1 •
+ 5— 2 OOSf/tf
|r (fr-08in(m+l)*(ft- j (ir-t)mn{m-l)tcu\
1.1 ^1 1 r ^ ^ 1
■■S + l<5<»'+S- 2 cos HI* >-= y
11 * cosms
5 + 70081- 2
2^4"^' ^,(m-l)(m+l)-
The required series is therefore
It will be observed that it is only for values of « between and w that the sum of this
series is proved to be —^ sins; thus for instance when « has a value between and — ir,
the sum of the series is not ^^^— sin^ but - ^5— sin g; when « has a value between ir
2 2
and 2ir, the sum of the series happens to be again — ^— sin s, but this must be regarded
as a mere coincidence arising from the special function considered, and not from the
general theorem.
Example 2. To expand ^-g— - ^ * «^« series, valid when 0<«<ir.
We have
— ^ — - ■■ — 2 sm 9IU I — ^-5 — •' am nUat
o n fli*i Jo o
■■ 2 smini J -^— ; — ^ BinnUdt.
Mai
'io-4
r«(ir-20ffln««n ^ 1 /■» . ,j.
Therefore
2m«
vziw'-t) . sinSf . sin5f .
140 THE PB0CES8ES OF ANALYSIS. [CHAP. VTI.
Here again the sum of the series is -^ — ^ only when z lies between and n. Thus
when s lies between «• and 2»r, the sum of the series is — ~ — . The sum of the
o
series for values of z beyond the limits and ir can be found at once from the equations
S (t)=» - S {" z) and S{z+2ir)^S(z)y where S(z) denotes the sum of the series.
Example 3. Prove that, when < « < ir,
yr (tr ~ 2z) ( w' + 2nz - 2^) cos3« cosSa
96 -cosa+-g^ +__+....
For when < « < «• we have
— ^ ^-i-jr^ i«- 2 cosm« I -> ^-^ "^cosm^cb
96 fTfwsi Jo 96
(integrating by parts) = 2 cosm« I sinm^cfe
■1=1 J 4^
(integrating by parts) = 2 cos me J ~-n- cos mtcft
m=i Jo 4m'
» /■» 1
(integrating by parts) = 2 cosm;; ) r— .sinmtcfo
mat y *^
« i-(-i)-
* — s — 2 — oosm*
m-i 2m*
. cos Zz . cos 5e
eoe f +— ^ + -^j~ + ... .
Excm/ple 4. Shew that for values of z between and ir, «** can be expanded in the
cosine series
and draw graphs of the function «** and of the simi of the series.
Example 5. Shew that for values of z between and ir, the function — ^^-^ — - can
be expanded in the cosine series
. cos Sz . cos 52
and draw graphs of the function -^-^ — < and of the sum of the series.
83. A Itemative proof of Fourier's theorem.
Another proof of Fourier's theorem, based on an entirely different set of ideas, is due
to Dirichlet*.
♦ CoUecUd W&rke, Vol. i. pp. 188—160.
83] FOURIER SERIEa 141
CoDsider first the sum of a limited number of terms of the series
00
where
a^^- lf{t)coafntcU (m«l, 2, 3, ...),
IT J
6^=i [^/(Osinm^cft (m-1, 2, 3, ...),
V J
and where z is supposed to be a real variable.
Since
V J
we have the sum to (2m +1) terms of the series expressed by the formula
^m-- /{i+coe(^-«)+cos2 (<-«)+. ..+oosm(<-«)}/(Oflfe
^ /'2»sin(2m+i;
h sm —
«r j , sin ^ J \ -^ I
2
=1 p5£(??^V(^+2^dd
fT J sm ^ -^ ^ '
1 p- Bin(2».+l)0
IT jo sm^ -^ ^ '
We have therefore to investigate the value to which integrals of this class tend as m
tends to infinity. Consider in general the value to which
/./*«£*!^(.)&
y sm «
tends when h^ supposed to be an odd integer, increases without limit.
First suppose 0<A<5 , and suppose that, for values of z within this range, ^ {z) is
continuous and positive, and that <^ {£) continually decreases as z increases.
Let -T- be the greatest multiple of t in A, so -r- < A < r+1 t .
142 THE PROCESSES OF ANALYSIS. [CHAP. Vn.
Then
» fw (n+l)ir f»
Now write
k
nw
k
+ ...+ /
•*
(r-D-
k
k
Binkz
ainz
n»
f * sin
\kz
'it\fl»
so U»'
ir
f * siniy - /nir , \ , , nv
I . /n7 \ »(,T-^y;^' where y=.--p.
The integrand in this last integral is clearly positive throughout the range of integration,
and Uj^ is therefore positive. Moreover, under the suppositions already stated, the
quantity
decreases as n increases, and it therefore follows that u^ decreases as n increases.
Also the well-known theorem of Mean Value shews that tin can be represented in the
form
where
ir
fk ainh/ .
and Pn-^f^+^K
S being some quantity between and r. Clearly y^ is positive, and decreases as n
increases.
Now we can write
Jrw smz ^^ *^
k
where J'=Wo-^i+t«|-Ws+...+(-iy"*Mr-i.
■
Since u^ is always positive, and decreases as n increases, we have
r-1
where m is any number less than —^ .
This gives
< "OPO- («'l-^l)Pt- ("8- »'4)P4- — -("jm-l - O Plm
-'l(Pl -P8)-»'8 (Ps-Pi)- — - "jm-lCPlm-i-Ptm).
83] FOURIER SERIES. 143
As p^ decreases with increase of n, the terms in the last line are negative, and can
be removed without affecting the inequality.
Thus
•^< ^oPto - ("l - "l) Ps - el's - •'4) P4 - • • • - (•'jm - 1 - I'm.) Ptm
<''0P0-(»'l-»'8)Plin-(»'8-^)Ptm-...-(''lm-l-Vjm)Pfm
-(»'l-''8)(Pt-Plm)-(»'3-l'4)(p4-Plm)-...-(»'jm-8-«'jm-8)(p2m-2-p2m).
The terms in the last line are again negative and can be removed. Thus
•^< "0 (PO - Pjm) + (''O - "1 + ''8 - • . • + ''Sm) Ptm.
We also have clearly
J'>tio-tii+w,- ... - 1^.1,
or •^>«'oPo-''iPi+''8P2- — -^jm+iPiw+i,
which in the same way gives
•^>Pfm (''0- 'l + ^S" ••• ~ "Jm+i)-
Thus J" is intermediate in value between the quantities
and Pim(''o-''i+*'2- — "-''8m+i).
Now let k become infinitely great, and let the quantity m likewise become infinitely
great, but in such a way that t tends to the limit zero. Then the quantities pQ and pi^i
tend to the limit ^ (0) ; and the quantity
(am-f 1) V
f * sinitv J
or / -: — '- dv
Jo smy ^
mm+Dir sint .
or 1 ^ cU
Jo 7 • *
•^ it; sin T
k
can, since k is infinitely large compared with m, be replaced by
/
(2in+l)»8in t ,,
*
and this, when m becomes infinitely great, tends to the limit ^ .
We see therefore that J is intermediate in value between two quantities, each of which
tends to the same limit, namely ^ ^ (0). J therefore tends to the limit q<I>(0); and
therefore /, which difiers from J only by a vanishing integral, likewise tends to the limit
Q<l>(0) as k becomes infinitely great. This result may be called Dirichle^i lenvma. To
complete the lemma, however, it will be necessary to shew that it is still true when a
number of the restrictions imposed on ^ {z) are removed.
(1) It was assumed that if> (z) was positive and steadily decreasing throughout the
range.
(a) Suppose that ^ (z) is constant This does not invalidate any of the preceding
proof, so the theorem still holds if ^ (1) is constant
144 THE PBOGBSSBS OF ANALT8I& [CHAP. Yll.
O) Suppose that ^ (z) is negative, or partly positive and partly n^^ative, but still
steadily decreasing; then choose a constant o so that c+^ (e) is positive through the range;
then the theorem applies both to o and to o+^ («) and therefore on subtraction to (i)
alone.
(y) Suppose that <f> (z) increases steadily throughout the range. Then the theorem
is true for { — <^ (z)} and therefore for <^ (z).
Therefore the theorem is still true if <^ (z) is ftmte^ continuous, and steadily increases
or decreases throughout the range.
(2) Instead of taking the integral between and A, take it between g and A, where
0<g<h^-^. We assume that the value of (t>{z) is only known for values of z from
^ to A.
Take a new function ^ («), defined as being equal to <^ (g), a constant, for values of z
from to ^, said equal to (z) for values of z from gU> L Then the theorem holds for 0^ («).
Also
^* /la* <^i(«)<&-|*i(0)=|<^(!7),
and
Therefore
by subtraction.
(3) Now assume there are a limited number n of maxima and minima within the
range to A.
Let them be at the values Oi, a^, ... 0^, of s. Then
On applying the theorem to each of these integrals in succession, it is clear that the
theorem holds for the whole integral.
Therefore the theorem is still true if <f) {z) is finite, continuous, and has not more than
a limited number of maxima and minima within the range.
It must be noted that these conditions still exclude such functions as e.g. (^-c) sin —
z^ c
where 0<c<A.
IT
(4) We shall now no longer restrict A to be less than 5- . Take < A ^ ir.
Then(a)let^<A<tr.
WriteA=tr-A', where < A' < | . Then
Lamit /=* / -: <p («) dz + I —. — <f> (z) dz,
j^moo J sm z J ^ SID z
83] FOURIER SERIES. 145
Writing z^n-Cin the latter integral, we have
ir »
Limit /« P2iji^*Wcir+ P«^!^*;0(.-f)eif.
Since <l>{fr-() satisfies the conditions stated, we see that when A' > the second integral
is zero.
^ Therefore Limit /- ^ <^ (0).
O) Let As TT. Then all the above reasoning applies, except that now A' 1*0, so
Limit/«|^(0)+f<^(ir),
which, in order to guard against uncertainty in the case in which the function is
discontinuous at and ir, is often written
where ff is a vanishing positive quantity.
(5) Next, suppose that the function <f) (z) within the range has a finite number of dis-
continuities, in the form of abrupt but finite changes of value. Divide the range into
various portions, so that each of them ends at one discontinuity and begins at the next,
and divide each of these into others each beginning and ending at a point of stationary
value. The above theorems apply to each of the portions, and therefore each integral is
zero except the first, which is equal to o^(<)) ^^d possibly the last, which when
h^n has the value o^Ctt-c).
(6) Finally, consider a function <t>{z) which becomes infinite for z^se, but in such
a way that the value of / ^ (z) dz tends to a definite limit as z approaches c from either
lower or greater values.
Then
where c is a small positive quantity.
In the second integral, a quantity ( can be chosen intermediate between c and c- c, such
sin JkC f^
that the integral is equal to -^— ^ / ^ («) c& ; on taking e small this vanishes ; and
sin C y c - •
similarly the third integral is zero.
On making k infinitely large, the fourth integral tends to zero. Therefore the theorem
holds in this case also.
(7) Thus we have, summarising the results obtained, the theorem that the Umit when
jk tends to infinity of I ^^ <l>{z)dzi8^<l){€)ifO<h<Viandis
Jo sm z 2
W. A. 10
146 THE PROCESSES OF ANALYSIS. [CHAP. Vll.
if h^it ; xohere € is a vanvUhing positive quantity ; provided thai <ft{z) is every where finite,
and has only a limited number of finite diecontinuitiee and maxima and minima between
the values and h of the variable z ; and this is still true if <f> {t) has a limited number of
singularities of specified type, namely such that I <f>{z)dz is finite.
This result may be called Dirichlefs lemma, the conditions just stated being referred
to as Dirichlefs conditions.
We can now return to the expansion which was found to represent the sum of the first
(2m + 1) terms of the Fourier series.
We had iS'w=/i+/„
X. r 1 /*'''«sin(2wi + l)tf ., , ^A^ jA
where /.--j^ ^__i_y(,+2tf)rf^,
1 r*«n(2^)
^ n Jo am tf -^ ^ ^
If 0<;r <29r, and/(j;) satisfies Dirichlet's conditions, we have hj Dirichlet's lemma
Limit /i— i/(«+€),
and
Limit /a=i/(2-€),
and so
Limit 5„-i {/(^+,) +/(«_,)}.
It z^O, we have
Limit /j = i {/(f) +/(27r - c)}, Limit 7,-0,
and so
Limit &,=:i {/(c) +/(27r - f )}.
If «=2ir, we have
Limit /i=0, Limit /,-*{/(«) +/(2ir-€)},
and so
Limit 5„=i{/(«)+/(2,r-€)}.
Thus we finally arrive at Fourier's theorem, namely that the sum to infinity of the
series
cio+ S (a^cos9iu+6mSinnu)
is f{z) at points z for which f is continuous, and is the arithmetic mean of the two values
off{z) at points z for which f is discontinuous : it being assumed that f{z) satisfies Diriohlet's
conditions.
Example, Prove that in the limit when n becomes infinitely great
/'•8in(2n + l)0 .^,. , ...
a being a real positive constant.
(Cambridge Mathematical Tripos, Part IL, 1894.)
84] FOURIER SERIES. 147
84 Nature of the convergence of a Fourier series.
The proofs of Fourier's theorem which have been given establish the result
only for the case in which the sequence of the terms in the series
2 (dm cos mz + bm sin mz)
is that in which m takes the orderly succession of values 1, 2, 3, 4, ... .
The question now arises whether the order of succession of the terms can
be deranged without affecting the value of the sum of the series ; in other
words, we have proved that the expansion of a function by Fourier's theorem
is a convergent series : we want to find whether it is absolutely convergent, or
only semi-convergent. The question has also to be considered whether the
series is uniformly convergent or non-unifomdy convergent in the neighbour-
hood of a given value of z.
We shall first shew by considering special cases that there is no geneml
answer to these questions.
Consider the series
sin i^— -sin 2^ + ssin3-«f— ... ,
which represents - z when Okzktt, and ^z^tr when ttkz <2Tr; this series
is semi-convergent for all real values of z, since sin n.^ is finite for all values
of n when z is real, and so the modulus of the general term bears a finite
ratio to the general term of the divergent series
In this series, therefore, the value of the sum will be modified if the order of
succession of the terms is changed.
Moreover, we can shew that the series is non-uniformly convergent at its
discontinuity tt. For the sum of the first n terms is
sin 2z . (- 1)***"* sin nz
smz 5 I-...H ,
2 n
or
/ (cos e - cos 2e + ...+(-!)'*"* cos 7rf)cie,
Jo
[* fl (- !)**-» co8 (n + l)^-hcosnel ,
Jo [2 "^ ~~2 nr^^^t 'J ^^-
The term I ^ dz represents the sum of the whole series ; so the remainder
Jo 2
after n terms, when — tt < z < tt, is
or
cosf»+ gj t
iin«(-ir'| ^ ^ dt.
2C08^
- 1)"-'
io
10—2
148 THE PROCESSES OP ANALYSia [CHAP. VII.
Writing <r = 7r — i;, ^■btt — w, this can be written
i2n=-|
sin (n+ gj u
dtL
28in^
Write (n + i j u « I'. The equation becomes
Sin V
dv.
However great n may be taken, if 17 be taken so small that (^ + 5)^ is
infinitesimal, this int^;ral tends to — I or — -5 , and so is not infini-
tesimal It follows that the series is non-uniformly convergent in the vicinity
of -^ — TT.
Consider next the series
1 0.1
COS<r + — C08 3ir + r:C0s5<r + .,. ,
o' o'
which represents ^^ q ^ when Okzktt, and — — ^ when 7r<z<27r,
This series is absolutely convergent for all real values of 5, since the moduli
of its terms are less than the corresponding terms of the convergent series
1 1
1 + g, + gj + . . . .
In this series therefore the order of succession of the terms can be changed
in any way, without altering the value of the sum of the series ; and since
the comparison series is independent of z, the series is also tmi/orrnlj/
convergent for all real values of z.
Returning now to the general Fourier series, we can discover the nature
of the convergence by a. consideration of the coeflBcients in the series, which
can be made in the following way.
We have shewn that if
ao
then
1 f *»
a,» = - 1 f(t) cos mtdt
84] FOURIER SERIES. 149
Suppose that (as in most of the examples we have discussed) the range
0<z<2Tr can be divided into other ranges, say 0<z<ki, ki<z<k^, ,,,y
kn<z< 27r, which are such that in each of these smaller ranges f(z) is
an analytic function of j?,, regular iu the range. (f(z) will not necessarily be
the same analytic function in the different ranges.) Thus it f(z) has the
value J? for < J? < TT, and has the value --z for ir <z< 27r, we should have
w « 1 and ki = tt. Then
1 r *» ,1 /**« 1 f**
am=- I /(t)coBmtdt + - f(t)coamtdt-\' ... + - I f(t)co8mt(JU.
Each of these integrals can then be integrated by parts ; we thus obtain
[*» 1 J.... sin mf\ , r*« 1 ^ ... sin vitl .
1 r*« 1 f*«
/ f'(t)smmtdt I f'(t)sinmtdt^ ... ,
TrmJo ''rmJkf ^
or
where
il = ~ [sin mJe, {/(k, - 0) -/(A, + 0)} + sin mk, {/(k, - 0) --/(k, + 0)} + ...],
and where bm is the coefficient of sin mz in the Fourier expansion of/' (z) —
an expansion which will exist, since/' (z) is a function of the same character
as f(z)y though the terms of this expansion will not always be the derivates
of the corresponding terms of the Fourier series {ot/{z).
Similarly
6 =:? + ^'
m m
where
-B«--[-/(+0) + cosmA;j{/(*,-0)-/(A^ + 0)} + co8mti{/(ifc,-0)
TT
-/(A, + 0)} + ...+/ (27r-0)],
and where a^' is the coefficient of cos mz in the Fourier expansion o{f'(z).
In the same way we have
m m
where
il' = ^ [sin 7n*i (/' (jfei - 0) VX*i + 0)} + sin TnJfc, {/' (ifc,- 0) V' (*«+ 0)}
150 THE PROCESSES OP ANALYSIS. [CHAP. VIL
and
, , B^ .dm
where
dm" and 6m ' being the coeflficients of cos mz and sin mz respectively in the
Fourier expansion of/'' {z).
Thus
A B! a^
//
, B A' bm"
6w =» - + — 1 r .
The conditions for the absolute convergence of the Fourier expansion of
f{z) are therefore expressed by the equations
il-0, 5 = 0;
for if these equations are satisfied, we have
a„ = .^?l±^'and6, ^'"^
m* *" m*
and the terms of the Fourier series are comparable with those of the con-
vergent series
1+1+1+1+
Now in order that we may have ^ = 0, B = 0, for all values of m we
must have
/(*.-0)=/(A, + 0),
/(Ar, - 0) =/(*, + 0).
That is to say, if a Fourier series is absolutely convergent for aM real values
of z, the fwnction represented by the series has no discontinuities, a/nd has
the same value at z^O as at z^ 27r.
If these conditions are satisfied the Fourier series is not only absolvlely,
but is also vmformly convergent For its coefficients a^ and 6m are in this
case of the order — ; , and so the series of constants
^ol + jchl + 16, ! + |a,| + |6, 1 +...
85] FOURIER SERIES. 151
converges ; but the moduli of the terms of the Fourier series are less than the
corresponding terms of this series, and consequently the Fourier series is
uniformly convergent for all real values of z.
Example 1. Shew that in general, when the Fourier series converges only for real
values of t^ the quantities a^ and h^^ can be expanded in infinite series of the form
of which the terms
m w' m' tn*
A B .B A'
5 and - + — ,
found above are the initial terms ; but that when the Fourier series converges within a belt
of finite breadth in the «-plane, all the coefficients c^) c,, C3, ... vanish, and this expansion
becomes illusory.
Example 2. Let /(«) be a function of «, which is regular for all real values of z
between ««=0 and «=7r, and which is zero at «=0 and z^rr. Prove that if /(«) is
expanded in a sine series, valid between z=0 and z^fry the series will be absolutely and
uniformly convergent for all real values of z.
Example 3. f{z) is a function of z which is regular for all real values of z between
and TT. Prove that if it is expanded in a cosine series, valid between zs=:0 and z^n, the
series will be absolutely and uniformly convergent for all real values of z,
86. Determination of points of discontinuity.
The expressions for dm, and 6^ which haye been found in the last
paragraph can be applied to determine the points at which the sum of a
given Fourier series is discontinuous. This can best be shewn by an
example.
Example, Let it be required to determine the places at which the sum of the series
sin £+} sin 32+i sin 5^+ .••
is discontinuous.
For this series we have
1
«m-0,
, 1 - COS mit
*»'- 2m"
Comparing this with the formula found in the last paragraph, we have
^ = 0, 5=J-j006m7r,
Hence \i k^^ k^,,, are the places at which the analytic character of the sum is broken,
we have
0=^-1 [8mm^i{/(iti-0)-/(iti+0)}+sin mi-, {/(it,-0)-/(it,+0)H...].
Since this is true for all values of m, the quantities i{-|, it,, ... must be multiples of n ; but
162 THE PROCESSES OP ANALYSIS. [CHAP. VII.
there is only one multiple of n in the range 0<g<2ir, namely n- itself. So k^^v, and
^2) ^8> ••• ^o not exist Substituting /r|«s7r in the equation ^-■^— ^oosmfr, we have
J-icoswir- [-/(+0)+COSW7r{/(7r-0)-/(»r+0)}+/(2»r-0)].
Since this is true for all values of m, we have
i=-^{/(2ir-0)-/(+0)},
and -j=-l{/(^-0)-/(ir+0)}.
IT
This shews that there is a discontinuity at the point x^nt such that
/(t-o)-/(«-+o)-|, ,
and that
/(2,-0)-/(+0)=-|.
Example. Find the discontinuities in value of the siun of the series
8ini-^sin22+^sin47-}sin5«+^sin7«-|sin82+^sin I0z+,..,
86. The uniqueneaa of the Fourier expansion.
We have seen that it / (z) is a quantity depending on z, and satisfying
certain conditions as to finiteness, etc., then the series
00+ 2 (Oro COS m« + 6,n sin m;?),
1 r*'
where a^ « - I f(t) cos mt dt (m > 1),
1 r*'
6m=»- / f(t)s]Ximtdt,
'^-llj^'^^'-
has the sum/(^) when ^ -? ^ 27r, except at the isolated points at which /(z)
is discontinuous.
The question arises whether any other expansion
00
Co + 2 (Cm COS wu + dm sin mz)
of the same form exists, which also represents f{z) in the interval from
to 27r ; in other words, whether the Fourier expansion is uniqtte.
We may observe that it is certainly possible to have other trigonometrical expansions
of (say) the form
00+ 2
(omCOSy+ftnCOSyj
86] FOURIER SERIES. 153
which represent f{z) between and %ir\ for write ;?«2^, and oonsider a function ^(C))
which is such that <^(f)«/(2f) when 0<f<ir, and <^(0-fl^(f) when 7r<f<27r,
where g (() is anj other function. Then on expanding ^ (^) in a Fourier expansion of
the form
00+ 2 (o^^cosmf+ftncosmf),
this expansion represents /(«) when 0<;r < 2fr ; and clearly by choosing the function g (()
in di£ferent ways an infinite number of such expansions can be obtained.
The question now at issue is, whether other series proceeding in sines and cosines of
integral multiples of z exist, which differ from Fourier's expansion and yet represent f{z)
between and 29r.
If it were possible to have a distinct expansion
. 00
/(^) = c^-f 2 (c» COS m* + cijn sin wiir),
m-l
then on subtracting this iix>m the Fourier expansion we should have an
expansion
(ao-Co)+ ^ {(a« - Cm) cos T/MT + (6,» - cim) sin 7?Mr}
whose sum is zero for all values of z between and 2ir, except possibly
a certain finite number of values (namely the discontinuities).
The investigation therefore turns on the question whether it is possible
for such an expansion as this last to exist. We shall shew that it cannot
exist, and that consequently the Fourier expansion is unique*.
Let' -Ao^gOo,
-^m =■ c^m cos m-? + 6m sin mr (m> 1);
and let
2 = -4.0 + Ai + ... + ilm + •••
be a convergent (not necessarily absolutely convergent) series for values of z
from to 27r, so that the limit of a^ and hn is zero forn:* oo ; and suppose
that (except at certain exceptional points) its sum is zero.
Then the series
ir ^<r;=sulo^ -ill- -^ -...-— -...
converges absolutely and uniformly for this range of values of z, as is seen
by comparing it with the series S — .
We shall first establish a lemma due to Riemannf , which may be stated
thus:
* The proof is due to G. Cantor, Journal fUr Math, Lxzn.
t Collected Worki, p. 218.
154 THE PROCESSES OF ANALYSIS. [CHAP. YU.
The quantity
jy F(z + 2a) -¥ F (z - 2a) - 2F(z)
^^ 4?
tends to the limit /(z) as a tends to zero, if at z the series X converges to the
sumf(z).
For the term involving On in J2 is
— -r^i (^ ^^^ n (ir -I- 2a) + a« cos n (-«? — 2a) - 2a^ cos nz],
On cos nz sin' na
or T-z ,
nV
J . ., , ^, ^ . , . I . bn Bin nz sin* na,
and similarly the term involving bn is -^ r^ .
As F(z) converges absolutely, we can rearrange the order of the terms,
and so can write
n . . /sinoV . A /sin2a\*
Now considering the series 1, we can write
say, where z being given, and any small quantity S being assigned at will, we
shall have | Cn | < S for values of n > some integer m.
Now An ■* «n+i — en for all values of n.
Therefore substituting, we have
Divide the series on the right-hand side of this equation into three parts,
for which respectively
(1) l<n<m,
(2) m + 1 < n < «, where s is the greatest integer in - ,
(3) « + 1 < 71.
The first part consists of a finite number of terms, each tending to zero
as a tends to zero, so the first part is zero.
Considering next the second part, the quantities are of the
form where <a<w; this quantity decreases as a increases from
to TT, so the sum of the moduli of the terms in the second part is less than
^ f/sinmay .sin^a.*]
which tends to zero when S tends to zero.
86] FOURIER SERIES. 155
Considering next the third part, we can write the nth term in the. form
Ksinn — lay /sinn— laVl «» /• . 7 • • \
or
sin* w — 1 a P 1 _ ll _ sin 2n — la sin o
^« a* [—J. n«J *** nV '
80, as I €n I < S, its modulus is less than
Thus the whole sum of the terms in the third part
5 1 sr_i j_ 1
8 r (ic 8 8 8 .8
which is ultimately zero. Therefore the three parts of the infinite series
in R are all zero ; and thus 22 '^fi^) in the limit ; which establishes Riemann's
lemma.
Next, we shall establish another lemma, due to Schwartz *, which may be
stated as follows : If a and b are two of the exceptional points, so that between
B^a and z^b the series 2 converges to the sum zero, then F{z) is a linear
function of z betu)een these values.
For assume that a is less than b, and introduce a function 4> (z\ defined by
4,{z)~0^F(z)-F(a)-l-^^{F{b)-F(a)\^-^{s^aHb-z).
where ^s 1 and h is any constant
Then substituting in the result of Riemann's lemma, we have
Therefore (^ + a) + ^ (^ - a) — 2^ (z) is positive when a is very small,
whatever be the value of z.
Now ^(a)«0 and ^(6) = 0. Also ^(z) is continuous, since F(z) is
uniformly convergent, and consequently continuous. Therefore if <f> (z) can
be positive between the values a and b of z, it will have a maximum ;
let this occur at the value c of z,
* Quoted by G. Cantor, Joumaifiir Math. Lxxn.
156 THE PROCESSES OP ANALYSIS. [CHAP. VII.
Then when a is small, we have
<t>(c + a)-4>(c)<0, and <^(c-o)- <^(c)<0.
Adding these relations, we see that the condition just found is violated, and
so <f) {z) can not be positive at all within the range.
Again, take h small. Choose ^ » j: 1, so choosing the sign that the first
term ^ [i^ (x:) — . . .] is positive. Then ^ {z) is clearly positive, if this first term
is not zero.
But j> {z) is not positive ; and thus we must have
Therefore F{z) ia ^ linear function of z, which establishes Schwartz's
lemma.
We see then that the curve y = F(z) represents a series of straight lines,
the beginning and end of each line corresponding to an exceptional point ;
and as F{z), being uniformly convergent, is a continuous function of z, these
lines must form parts of a polygon.
But by Riemann's lemma
limit ^i' + '0-f(') F(z-a)-F(z) ^Q
Now the first of these fractions gives the inclination of the earlier side of
the polygon at a vertex and the second of the later ; therefore the two sides
are continuous in direction, so the equation y=^F(z) represents a single line.
If then we write F (z) ^cz-^ c\ it follows that c and c' have the same values
throughout the range. Thus
and therefore
A ^ A ^tl I
the right-hand side of this equation being periodic, with period 27r.
The left-hand side of this equation must therefore be periodic, with period
27r. Thus we have
4^ = 0, c = 0,
n h
and — c's=ili-H...-H-r cosn^4-~sinn5-H....
nr n'
Now the right-hand side of this equation converges uniformly, so we can
86] FOURIER SERIES. 157
multiply the equation by cos nz, sin nz, respectively, and integrate. This
gives
TT ~ =« - c' I COS nz4^ « 0,
^ J
b [^
and TT -4 « — c' I sin nzdz « 0.
n« Jo
Therefore the a's and b's vanish, so all the coefficients in S vanish ; which
establishes the result that the Fourier expansion is unique.
Miscellaneous Examples.
m
1. Obtain the expansioiis
(«) 1 — o — ^=H-rooe*+r«coe2f+... ,
^ * l-Srcoe^+r"
W 5log(l-2rooe«+r^— -rcoB«-jr*co8 2«-5r"ooe3;«-...,
(c) tan"** _ — r8in«+5r*8in2r+5r"8in3;«+...,
^ * l-rcoB« 8 3 '
(a) tan"*-= z-=r8m«+s*'^sui3«+rr*8m5*+...,
^ ' 1—1* 3 5 '
and shew that, when | r | < 1, they are oonvergent for all values of t in certain belts
parallel to the real axis in the ^plane.
2. Shew that the series
- sin — sin^ 8in(n+l) — sin |
where all the terms for which i; is a multiple of n are omitted, represents the greatest
integer contained in a, for all real values of z between and n.
3. Shew that the expansions
^l0g(2<X)8|j
cost - 5C0s2;;+xC0S32...
and
- log(2 8in rj= -co8«-5 008 2;j-s008 3«...
are valid for all real values of «, except mtdtiples of ir.
4. Obtain the expansion
• (-l)*^co8fn« , «., /„ '\ . ' / • o . • \
and find the range of values of t for which it is applicable.
(Trinity College, 1898.)
158
THE PROCESSES OF ANALYSIS.
[chap. VII.
5. Let n be an integer ^ 2, and let XnX^^... x^^i be quantities satisfying the conditions
0<a?i<ar,<...<ar,i_j<l,
and write Xq=0, ^n**!*
Let 0^, 0], c^t ... c^.i be real arbitrary constants and let a function <f> {a) be defined by
the equalities
<^(ar)=c^+(Ji+..,+c„ for ar,<a?<it:,+i (i—O, 1, 2, ...n-l),
<^(x)-Coforx«Xo,
Cm
<^(^)-Co+C|+...+c^i+^, for x^Sf
Shew that
for X^Xn*
(*-l, 2, ... n-1),
2 Mai
for 0<a?<l,
and
*jWjt(i2.5+ i «..
where the coefficients a^ and 6^,^ are given by
11-1
ao«2 2 (v(l-.JPr),
o,^" 2 (V8in2m9r^r>
wiir
for m'^ly
1 »-*
6«=--- 2 (v(l-coe2mira?r)
for m^l.
(Beiger.)
6. Shew that between the values - v and + ir of t the following expansions hold :
3 sin 3s
mximz
2 . / sini 2 sin 2s 3 sin 3s \
2 .
COS msB- sin imr
(_1_ rncos* fnoos2s moos3s \
2^'^F^^*'' 2«-m« ^ 3«-m« " "7'
-Hg"*^ ^2 / 1 moosg
m cos 2s wicos3 s
2*+«? " 3*+wi
7. Obtain the expansions
5 sin/i(f-fmtr)
■-00 s+m*r
sin(2n-|-l)s
and
• cos/*(s+mfr)
2
IB — 00
s+mir
Sin I
,sin2nscoti
' oos(2n + l) s
sins
cos 2ns cot z
(2n</i<2n+2)
0A-2nX
(2n</i<n+2)
0»-2nX
MISC. EXS.] FOURIKB SERIES. 159
lip and q are positive integers, shew that
^ Bin (am -k-p) — ^
2 i I »~8m — ^- cot^--,
».— ym+jt) ^ ^ ^ '
cos(^+p)?^
* ^^ '^ a IT 2npir .^pn
2 * aa — cos ^ COt-^— . .
m— «D ^-m+jt) ^ ^ q
8. Prove that the locus represented by
2 - — I — sinitrsinnyaO
is two systems of lines at right angles, dividing the coordinate plane into squares of
area nK
(Cambridge Mathematical Tripos, Part I., 1895.)
9. If m is an integer, shew that
^ o 1.3.5...(2m-l) fl , »i o . m(m-l) .
2. 4. 6... 2m (2 m + 1 (fn + l)(m+2)
m(m-l)(m-2) 1
■*'(m+l)(i»+2)(nH-3)*^'^'^-7
(a terminating series),
^, 4 2.4.6...(2m-2) fl ^2i»-l ^ r2m-l)(2m-3) ^ . )
^^'^^■^ 1.3.5...(2m-l) i2-^2iM:i^^-^ (2m-hl^^
(an infinite series).
Shew also that
, 4/ oos3s.co8 5;s oos7«.oos9« \
and
cos
1 - / oosg , ooe3g ^ oosSg cos7i co6 9f \
' *"irV1.3"^1.3.6 3.5.7"*"6.7.9'"7.9.1l"^*'7'
10. A point moves in a straight line with a velocity shich is initially ti, and which
receives constant increments, each equal to ti^ at equal intervals r. Prove that the velocity
at any time t after the beginning of the motion is
u ^ ut u • 1 . 2mirt
- + — +- 2 -am — ,
and that the distance traversed is
trf ,^ . . . Mr VT • 1 2mirt
2;('+") + i2-2,ri,.!,JSS^-T-'
11. Shew that
• / .« . sinarir • (-l)*sin(a+2nvir)
sin(a + 2tMr9r)» 2 ^ ^— ^ ',
It .00 x—n
where n is the difference between the real quantity v (supposed not to be an odd multiple
of \) and the integer to which v is most nearly equal
(Cambridge Mathematical Tripos, Part II., 1896.)
160 THE PBOCfiSSES OF ANALYSIS. [CHAP. VIL
12. Let 9t be an integer > 3, and let ^o, ^|, ^2> <•• be an infinite aet of quantities, which
satisfy the conditions,
Let X be a real variable, and let s be the greatest integer contained in nx.
Shew that when x > 0,
2 ^r= o + 2 {(hnCoa2inirx+b^sin2mnx)f
if r is not a multiple of - ;
2
but
2 ^r-o "?+ 2 (aMOOs2fnir^H-6m8in2in9r^,
2 2 masl
if r is a multiple of - ; the coefficients a^ and h^ being determined by the formulae
fi
^" " •n^ 2 ^,.8in -— (m^ 1),
1^ n— 1 2fitr9r
^"^ — 2 9r<^—— («i>l). (Berger.)
13. Let ^ be a real variable between and 1, and let n be an integer > 5, of the form
4m+ 1, where tn is an int^;er.
Let E (a) denote the greatest integer contained in a.
Shew that
(-1) v»/ + (— 1) v « / = _ + _ 2 —tan cos2m9r^,
itx\a not a multiple of - ;
but
. nnx . ooBnnx 2.2 • 1 ^ 2mfr
sin ^ + — ' — ■« - + _ 2 — tan cos 2mirx.
if 0? is a multiple of - . (Berger.)
14. Let a; be a real variable between and 1, and let n be an odd number ^ 3.
Shew that
( — 1)*«- + — 2 —tan — cos29n9rar,
if 0? is not a multiple of - , where < is the greatest integer contained in nx\ but
tv
MISO. £XS.] FOURIER SERIES. 161
0=-+— 2 -tan — coa2inirx.
n n si3| m n
1
if 4? is a multiple of - . (Berger.)
15. Let X denote a real variable between and 1, and let 9i be an integer > 3 ; further,
let E(a) be the greatest integer contained in a. Shew that
n wi IT m»i f^ n
if ^ is not a. multiple of - ; but
. ^1 (n-l)(n-2)^l • 1 .mir ^
tu;*— Twr+s" ^ ^ + - 2 — cot - co8 2m»rj?,
as on . IT mmi m n
1
if ;p is a multiple of - . (Berger.) '
tit
16. Assuming the possibility of expanding f(x) in a series of the form lAjg sin kx,
where 1; is a root of the equation i&co8ai;+6sinai:=0, and the summation is extended
to all positive roots of this equation, determine the constants Aj^,
(Cambridge Mathematical Tripos, Part I., 1898.)
17. If
shew that
6*-l ftsO ^J
. C0s4Yr^ . C0s6irJ7 . , ,v. , 2**~*ir** „ , .
cos2ir^+— 2^- +— 3s^+... = (-l)"-^-2;jj- V^(x\
ainSiixl""^'"^ I »^^^^^ | _/_n«+i?!!!!:^V (x)
18. If
shew that
(Cambridge Mathematical Tripos, Part II., 1896.)
/(x) = ^Oq + Ox cos a;+ a, cos 2^ + . . . ,
If
shew that
a^ss" I f{x)ooanxtsji^x -.
If J ^
<^ (j*)=6| sin ^+62sin 2a?+ ... )
b^xM - I <l}{x)axinx tan ^a? — . (Beau.)
19. Prove that the series 2ii»sin - — ,
1 <*
where ^n^zl sm -— -/(t;) cfv,
is equal tof(x) for any value of x lying between and a about which f(x) is continuous.
W. A. 11
162 THE PROCESSES OF ANALTSia [CHAP. VIL
If /(O), /(a) are the limits of /(«), /{a^^tX when the positive quantity c diminishes to
zero, and if /(:r) has sudden increases of value A, k, corresponding to the values a, /3, ... of x,
the limit for 7»= ao of nA^ can be written in the form
i|/(0)-(-l)./(«)+Acoe^+*co8^ + ...}.
Shew that the series
sin3a;+- Bin9x+-sin 15:t7+...-2 (sina?+- sin3:F-|--sin 5jf-|-...]
^3^3/ . 1 . K . 1 • ^ 1 • 11 ^ 1
H -jsmd?- r|8ln5ar+=jSm7A•-_-T^smll4?+..,V
ha8 the limit -^n* when Xy lying between and 9r, approaches indefinitely near to one or
other value, and that it has sudden changes of value -^ir and H-^fr corresponding to the
values i n and § n- of :r.
(Cambridge Mathematical Tripos, Part I., 1893.)
20. If, for all real values of x,
F{x) = AQ+AiCosx+AiCOB2x+A^co&3a;+,,,,
then
cos(^jF(^)d^= / XF(x)dx,
where
£r= il^ -I- Jj cos IT + il J cos 4tr + -ij cos 9tt? + . . . ,
F=4i sin i^+^ij sin 4M;+il3 sin 9tr+ ... ,
-, X^ ^^ An^^-X^ ira? , ^ 4(2»r)«+a:» 2ir:r .
^=COS -;-+2C03 COS h2C0S — — J- COS + ...
Aw Aw w Aw w
4(2n»r)«+a:» ^nirx
+ 2 COS -^ — 7 cos .
Aw w
Prove these formulae, and thence deduce the result
{U+ V) (^y-l^(0)+^Mco8^^'?+/'(2«-)oo8 ?*;"+...
2ir
where to^-r- ^ k being a positive integer. When k is even, the last term of each series
involves F{ikw) and is to be multiplied by ^; when k is uneven, the last term involves
F{i{k-l)io),
(Cambridge Mathematical Tripos, Part II., 1896.)
CHAPTER Vni.
Asymptotic Expansions.
87. Simple example of an asymptotic expansion.
Consider the function
• el^Ht
J 3
t
where x is real and positive, and the path of integration is the real axis in
the ^-plane.
Integrating by parts, we have
and by repeated integration by parts, we obtain
^/ \ 1 1.2! . (-!)»-' (»-l)! / ,w , r«^*d«
In connexion with the function f{x\ we therefore consider the series
1 1^2! ^ (-!).-.(„ -1)1
X oc^ a? '*' x^
We shall denote this series by S, and shall write
1 12! (-l)nn! _
X ^■*"a;» •••"^ (^"^^ "'^~*
n-1
The ratio of the nth term of the series & to the (n — l)th term is :
^ ' X
for values of n greater than 1 + a?, this is greater than unity. The series 8 is
iherefo7'e divergent for aU values of x. In spite of this, however, the series
can under certain circumstances be used for the calculation of f{x); this can
be seen in the following way.
11—2
164 THE PROCESSES OF ANALYSIS. [CHAP. YIII.
Take any definite value for the number n, and calculate the value of Sn^
We have
/(^)-Sfn-(-l)«+'(«+l)!/J'^f.
and therefore
\/(x)-8n\=(n + lV.(
J i
— ^ , since e*^ < 1 and t is positive^
<
^n+i-
For values of x which are suflBciently large, the right-hand member of
this equation is very small. Thus if we take x > 2n, we have
1
|/(^)-^n|<
2«+i n
1 >
which for large values of w is very small. It follows therefore that the value-
of the fwaction f(x) can be calcukUed with great accuracy for large values
of X, by talcing the sum of a finite number of terms of the series 8.
The series is on this account said to be an asymptotic expansion of the-
, function f{x). The precise definition of an asymptotic expansion will now^
be given.
88. Definition of an asymptotic expansion,
A divergent series
-^0 + — H"-^ +.••• + ^ + •••»
in which the sum of the (w + 1) first terms is Sn, is said to be an asymptotic
expansion of a function f{x), if the expression af^ \f{^) - Sn] tends to zero
as X (supposed for the present to be real and positive) increases indefinitely-
When this is the case, if a? is sufficiently great, we have
where € is very small ; and the error — committed in taking for f{x) the-
(n-f 1) first terms of the series is very small. This error is in fact infini-
tesimal compared with the error committed in taking for f{x) the n first,
terms of the series : for this latter error is
of" '
and € is in general infinitely small compared with iln + €.
88,89]
ASYMPTOTIC EXPANSIONS.
165
The definition which has just been given is due to Poincar^*. Special
asymptotic expansions had, however, been discovered and used in the
eighteenth century by Stirling, Maclaurin and Euler. Asymptotic expan-
sions are of great importance in the theory of Linear Differential Equations,
and in Dynamical Astronomy ; these applications are, however, outside the
scope of the present work, and for them reference may be made to Schle-
singer 8 Handbuch der Theorie der linearen Differentialgteichungen, and the
second volume of Poiucar6's Les M4thode8 Nouvdles de la Micanique^ Cileste.
The example discussed in the preceding article clearly satisfies the
definition just given : for
n\
and the right-hand member of this equation tends to zero as x tends to
infinity.
The term "asymptotic expansion" is sometimes used in a somewhat
wider sense ; if F, ^, and / are three functions of a?, and if a series
X or
is the as3anptotic expansion of the function
F '
we can say that the series
^ „. FA^ FA^
X
as"
is an asymptotic expansion of the function J,
For the sake of simplicity, we shall consider asymptotic expansions only in connexion
with real positive values of the argument. The theory for complex values of the argiunent
may be discussed by an extension of the analysis.
89. Another example of an asymptotic expansion.
As a second example, consider the function f{x), represented by the
series
.(1),
where c is a positive constant less than unity.
The ratio of the Arth terra of this series to the {k — l)th is less than
unity when k is large, except when a? is a negative integer, and conse-
♦ Acta Mathematical vin. (1886), pp. 295—344.
166 THE PROCESSES OF ANALYSIS. [CHAP. VIII.
quently the series converges for all values of x except negative integral
values. We shall confine our attention to positive values of x. We have,
when x>k,
1 I k k" 1^ h
= = + -^--.+ -7-....
X + k X (/^ s? SK^ a^
If, therefore, it were allowable to expand each fraction , in this way,
X T" fu
and to rearrange the series (1) according to descending powers of a?, we
should obtain the series /
t4"-^-4'^ w-
00
where ili= 2 c*; il3 = — 2 A;c*, etc.
But this procedure is not legitimate, and in fact the series (2) diverges.
We can, however, shew that the series (2) is an asymptotic expansion of
/(a?), which will enable us to calculate /(a?) for large values of x.
Forlet' s„ = =^ + ^» + ...+^'.
X a^ af^^^
and ^{/(^)-£f„} = LjL2_ 2 ^.
Now 2 r is finite, and so when x is infinitely great the right-hand
t— 1 X -h K
member is infinitesimal.
Therefore a?"{/(a?) — /Sn} tends to zero when x tends to infinity; and so
the series (2) is an asymptotic expansion of f(x).
Example. l{f(x)= I e^^^dt^ where x is supposed to be real and positive and the
path of integration is real, prove that the divergent series
\ 1 1.3 1.3.5
is the asymptotic expansion otf(x).
90] ASYMPTOTIC EXPANSIONS. 167
90. Multiplication of asymptotic expansions.
We shall now shew that two asymptotic expansions can be multiplied
together in the same way as ordinary sehes, the result being a new
asymptotic expansion.
For suppose that
X a^ '" af^
•^0 '• "T" "^ ^^"T •••4" _„■ > •••
are asymptotic expansions representing functions J{x) and J'{x) respectively,
and let 8^ and 8n be the sums of their (n + 1) first terms ; so that
Limit a?« (/ - 8n) = 0^
(1).
Limita;»(J-'-/S„') =
ap=300 ,
Form the product of the two series in the ordinary way ; let it be
and let Sn be the sum of its n first terms.
As 8n, 8n and 2» are simply polynomials in - , we have clearly
X
Limit «»(flf„S„'-S„) = (2).
gmto
Now by (1), we can write
'^ = ^« + ^'
where Limit e = 0, Limit e' = 0.
ee'
Then a!» (J J' - S„ 5«') = Sn'e + /S„€' + ^: .
X
The terms in the right-hand member tend to zero as x tends to infinity.
Hence
Limit a?«(JJ'-Sn/Sfn')=0..... (3).
«B00
168 THE PROCESSES OF ANALYSIS. [CHAP. VIIL
From (2) and (3) we have
Limit a^ {J J' - 2n) = 0,
X=oo
and therefore the series
X or
is the asymptotic expansion of the function JJ\
91. Integration of asymptotic expansions.
We shall now shew that it is permissible to integrate an asymptotic
expansion term by term, the resulting series being the asymptotic expansion
of the function represented by the original series.
For let the series
represent the function J(x) asymptotically, and let Sn denote the sum
Then, however small a real positive constant quantity € may be taken, it is
possible to choose x so large that
and therefore
J(<')-Sn\<^,
\r J{x)dx-{ Sndx\^r\J{x)-8n\dx
\j 9 J X I J »
^(n-l)«»-»'
and therefore the integrated series
a, ^2<r»+-+(n-l)«»-> + -
is the asymptotic expansion of the function
J{x) dx.
J a
X
On the other hand, it is not in general permissible to di£ferentiate an asymptotic
expansion.
92. Uniqueness of an asymptotic expansion.
A question naturally suggests itself, as to whether a given series can be
the asymptotic expansion of several distinct functions. The answer to this
91, 92] ASYMPTOTIC EXPANSIONa 169
is in the affirmative. To shew this, we first observe that there are
functions L{x) which are represented asymptotically by a series all of
whose terms are zero, i.e. functions such that
Limit x^L {x) = 0,
ivhatever n may be, when a; (supposed to be real and positive) increases
indefinitely. The function e~* is in fact such a function. The asymptotic
•expansion of a function J{x) is therefore also the a83anptotic expansion of
J{x)-¥L{x).
On the other hand, a function cannot be represented by more than one
-distinct asymptotic expansion for real positive values of x ; for if
X a^
-4o+— + —'+.. .
-and jBo + — +-?+...
X a^
«re two asymptotic expansions of the same function, then
U^i^(A4" + ...4--A-4....-|.).„,
which can only be if ul© = 5© ; -^i » -Bi, etc.
Important examples of asymptotic expansions will be discussed later, in connexion
with the Gkunma-function and the Bessel functions.
Miscellaneous Examples.
1. Shew that the series
1 1 ' 2 ' 3 '
- + —+ — + — -I-
is the asymptotic expansion of the function
/
dt
t
when X is real and positive.
2. Discuss the representation of the function
(where x is supposed real and positive, and <^ is an arbitrary function of its argument) by
means of the series
170 THE PROCESSES OF ANALYSIS [CHAP. VDI.
Shew that in certain cases (e.g. (f)(t)^e^ the series is absolutely convergent, and
represents f{x) for large positive values of j? ; but that in certain other cases the series is
the asymptotic expansion o{f(x).
3. Shew that the divergent series
I a-l (a -l)(a-2)
is the asymptotic ezpemsion of the function
«-• r*
log^j,
for large positive values of z,
4. Shew that the function
/(.)=/; (iog«-.iog(^)}^-
6-»»
has the asymptotic expansion "
where By^B^y ... are Bemoulirs numbers.
Shew also that/(j;) can be developed as an absolutely convergent series of the form
5. Shew that the function
has the asymptotic expansion
1.3.*.(2n-3)
00
PAET II.
THE TRANSCENDENTAL FUNCTIONS.
CHAPTER IX.
The gamma-function.
93. Definition of the Oamma-function : Euler*8 form.
Consider tlie infinite product
-n
n
This product clearly diverges if -e is a negative integer, for then one of
the denominator-factors vanishes. If z is not a negative integer, the product
will (§ 23) be absolutely convergent, provided the series
Jj.log(l + l)-log(l + i).
is absolutely convergent ; but since when n is large we have
the terras of this series ultimately bear a finite value to the terms of the
series
and therefore to the terms of the series 2 — , which is absolutely convergent.
The infinite product is therefore absolutely convergent for all values of z,
except negative integral values.
174 TRANSCENDENTAL FUNCTIONS. [CHAP. IX.
This product may be regarded as the definition of a new function of the
variable z\ we shall call it the Oammorfunction, and denote it by r(^),
so that
r{z)=-u
i.('-S"
n
This form of the function was fii-st given by Euler ; but the notation r(^)
is due to Legendre, who applied it in 1814 to an integral which will presently
be discussed, and which represents the Gamma-function in some case&
Excunple. Prove that
r(;?)= Limit -r-r -ryr/— — ^T^'• (Gauss.)
•«=«, «(«+I)...(«+n-l)
94. The Weierstrassian form for the Oamrnxi-fmiction.
Another form of the Gamma-function can be obtained as follows:
We have
, n+l
v{z)^- n^
*»-! 1 + 1
n
= iLimit «*><«'"'+« fi ^
Z fHsoo
"-('^9
= - Limit 6'{"*<~+"-'-«---»}n -21-.
1+-
n
Now ! + _+.. .+--.-log(m+l)= t (--log
n+1
n
= 1 ni-±-)i^
Joln-iw(n-fa?)j
Now the series 2 —. r is absolutely and uniformly convergent for
real values of a? between and 1, as is seen by comparing it with the series
* 1
n=l ^
94] THE GAMMA-FUNCTION. 175
hence as m increases, the right-hand member of this equation tends to the
limit
1 ( 00
io(n=iw(n+"^
which is finite, since the range of integration is finite and the sum of the
w
series 2 —7 r is finite. This limit is known as Euler's constant, and we
ii=in(n+a?)
shall denote it by 7, Its numerical value is
0-5772157. ...
Thus Limit ]i + h+... + log(m + l))- = 7,
z
1 * 6
and 80 r (^) = - e-^' II
or
^ n=i 2 , f^ '
n
}-^zey'U |fl + ^)^"*l
W n=l (V n) J
This form (due to Weierstrass) shews that yr7-\ is a regular function of z
for all values of z.
Example 1. Prove that
1^(1)= -y,
where y is Euler's constant.
For differentiating logarithmically the equation
and putting z=\ after the differentiations have been performed, we have
-r'(i)=i+y+i(^^-l).
or r'(l)«-y.
Example 2. Shew that
2 3 n Jo »
and hence that Euler's constant y is given by
' Jo «
Example 3. Shew that the infinite product
176 TBANSCENDENTAL FUNCTIONS. [CHAP. IX,
has the value
For nil — )««= n e*
i»»i \ z-¥nJ n=i n+z
*^* {n+z)e'H
5 (i+izf)
e n
n fl+-^tf-^
The numerator of this expression is Weierstrass' form of
1
(«-j?)6>^-'>r(a-a?)'
and the denominator is
1
ze^'riz)
Therefore the given expression has the value
e^zr (z)
96. The difference'equaiion satisfied by the Gamma-function,
We shall now shew that the function r(^) satisfies the difference-
equation
r(^ + i) = ^r(^).
We have
1 «(^ + 9
r(^+i) = -^n — 2i_
1 +
n
IVn-^l
z-^-ln^i n-^^ + l
n
00
1 +
71/ il V W/
u- — —= n
n+1 n
This is one of the most characteristic properties of the Gamma-function.
It follows that if 2r is a positive integer, we have
r(z) = (^-i)!
96,96] THE GAMMA-FUNCTION. 177
Example, Prove that
111'
e
r(r+i) r(z+2) ' r(«+3)
\z l!«+l'*"2!«+2 Sl^+a"*" •••J •
For consider the quantity
1 1 1
^'*"«(i5+l)'*"^(«+l)(z+2) **■••••
This can be expressed as the sum of a number of partial fractions, in the form
z ^«+l^**'^«+n^*'"
To find the coefficients a, multiply by {z-\-n) and put «= -n ; we thus obtain
1 fll 1 )(")*« ^
^"'(-i)"7i!r'*'i'*'r2"*"r70"^ "J "~w^ *
Therefore
z
But
2^«(2+l)^z(«+l)(z+2)^- ^\2 (0+1)1! (^+2) 2! -J
1 r(«+n+l)
«(z+i)...(«+«) r(2)
whence the required result follows.
96. Evaluation of a general cla^s of infinite products.
By means of the Gamma-function, it is possible to evaluate the general
class of infinite products of the form
00
n Un,
where v^ is any rational function of its index n.
For resolving Un into its factors with respect to w, we can write the infinite
product in the form
Yi j Q/t - ai)(n- g,) ... (n-aj
„«i ( (n-6i) ...(n-6j)
In order that this product may converge, it is clearly necessary that the
number of factors in the numerator may be the same as the number of
factors in the denominator; for otherwise the general term of the product
would not tend to the value unity as n tends to infinity.
w. A, 12
178 TRANSCENDENTAL FUNCTIONS. [CHAP. IX.
We have therefore A: = /, and can write (denoting the product by P)
n-il(n-6i) ... (n-6fc)j
For large values of n, this general term can be expanded in the form
('-3-('-?)('-r-('-r'
or 1 h terms m — r + . . . .
n n'
In order that the infinite product may be absolutely convergent, it is
therefore further necessary that
Oi + ... + ajb — 6i — . . . — 6t = 0.
We can therefore introduce a fiwtor
e ^
in the general term of the product, without altering its value ; and we thus
have
But n j(l--)e"U „/ - .^^ .
Therefore P= ^^^^dhlM (=*«) •_-.l»£(zi_M
a formula which expresses the general infinite product P in terms of the
Gamma-function.
Example 1. Prove that
•J" «(«_-H5>+f) ^r(a+i)^r(6+i)
,=i<a+«)(6 + *) "r(a + 6 + l) *
Example 2. Shew that
^(i-g)^i-g)...-{-r(--:»;^)r(-a^)...r(-a-iJ)}-S
where
2ir . . 2ir
a=cos hi Sin — .
n n
\
97, 98]
THE GAMMA-rUNCrriON.
179
97. Conneodon between the Gammorfunction and the circular functiovs.
We now proceed to establish another of the characteristic properties of
the Gamma-function, expressed by the equation
r(z)r(l-z)=^
IT
SUITTZ
We have
2(i—Z) „.i
(•^3
]^\«+l— «
W-f 1
1 •
n
1 •
n
Zil-Z) n^i
=i n ^
(•-9('-.-Tl)
'■(-5)
TT
Sin TTZ
which is the result stated.
Corollary. If we assign to z the special value g, this formula gives
|rg)('=..orr(i) = .*
98. The multiplication-theorem of Oauss and Legendre.
We shall next obtain the result
»-i
r(^)r (^ + 1) r [z+fj ... r (^ +^) = r (n^)(2,r) * «*-».
n
ni;
For let
0(^) =
r(^)r(^ + ^)...r(;r+
n-l
w
Then
<l>(z
nF (n^)
^rur+nf (^ + 1) F (^ + 1 + i) ... F (^+ 1 + ^^)
nF (n-2 + w)
(m + n - 1) (nz + » — 2) ... (nz)
^W
12—2
180 TRANSCENDENTAL FUNCTIONS. [CHAP. IX.
It follows from this that ^ (2) is a one- valued function of 2, with the period
unity ; and <f> (z) has no singularities when the real part of ^ is positive, since
=-7 — r is ever3rwhere regular ; it has therefore no singularity for any value
of z, and so by Liouville's theorem (§ 47) it is a constant.
Thus <j) (z) is equal to the value which it has when ^ = - ; which gives
The«f.„ *F^- {r (1) r (I - 1)) {r (?) r (1 - 1)}
(by § 97) = 1:11 ^ (2^^!:
Thus <l>(z) =
. TT . 27r . (n — l)7r n
sm - sm ... sin ^ —
n n n
«-i
or
r(^)r(^+^) ...r(^+'^) = r(n^)n*-~«(27r)
n-1
2
Example. If
shew that
Binp,nq)=n-'^ it(g,g)iK2g,g^ ... ^K>^- l)g, g} '
99. Expansions for the logarithmic derivates of the Gammorfunction.
We have
{r (z + !)}-» = ^^n (1 -I- ^ €~\
DiflFerentiating logarithmically, this gives
dlogr(^ + l) _ f ^^ 1 1 \
dz '^■^^(l(^+l)"^"2(^+2)"'"3(^ + 3)"^"y
Also
iogr(^ + i) = iog^ + iogr(^),
so
^iogr(.+i)-U^iogr(.).
99, 100] THE GAMMA-FUNCnON. 181
Therefore
^iogr(.) = l+*iogr(.+i)
1 d_( z
+ 1) "^'2(^ + 2) ■*■•••
_ 1 1 1
"■^«'*"(^ + l)>"^(^ + 2)«^--
These expansions are occasionally used in applications of the theory.
100. Heine's expression ofT{z)asa contour integral.
It has long been recognised that the Oamma-fiinction is intimately
connected with the theory of a large and important group of definite integrals ;
and in fact the function has frequently been defined by means of a definite
integral. We now proceed to consider various definite integrals in this
connexion, the most general of which is due to Heine and can be obtained
in the following way.
We have /, IV
i^('-3"
r(^) = - n
m
1 ^- 7n
Now if we express - 11 in partial fractions, we obtain
i n — = 1 (-i)" — — =—
^ m»i z-^m mao tn ! (n — m) \ z + m
Consider now the function
(- ^)-'.
This, when a is a complex quantity, may be defined as being equiva-
lent to
r«-i)iog(-»)
Now the logarithmic function is many-valued, since the value of the
function log (— w) is increased or decreased by 27ri when the variable x describes
a simple circuit round the point a? = 0. In order that the function (- a?)*"*
may have a unique value, we have therefore to select one of the different
determinations of log (-a;): and this may be done in the following way.
We first make the stipulation that the variable x is not to cross the real
axis at any point on the positive side of the origin ; this prevents x from
making circuits round the origin, and so makes each of the determinations of
log (— x) a single-valued function. Then we select, fix>m these determinations,
that one which makes log (— x) real when ^ is a real negative quantity. The
value of log (— x) being thus uniquely defined for every value of x, it follows
that the value of (—a:)*"* is likewise uniquely defined.
182 TRANSCENDENTAL FUNCTIONS. [CHAP. IX.
With these presuppositions, if (7 be any simple contour enclosing the origin
and cutting the real axis in the point x^l,we have clearly
f , , , , r (-a?)»l 6^ c-'*^ 2tsin7ra
Jc L(7 « J a a a
Therefore
iniLl^=,-.i-j(i+i)...(i+i)ri f (-rrv-.y
^m«i 1 , ^ 2sm7r^(V 1/ V w/j ««oJ(7 \mj^ '
+m-i^
1+-
(n + iyf (- a?y-i (1 - a?)** (ic.
2 sin TT-er ^ J c
Writing y = ti^ in this equality, we obtain
IV
1 liiL!j2l = _^
m
where 2) denotes any simple contour in the plane of the complex variable y^
enclosing the point y = 0, and cutting the real axis in the point y«n. If
now we make n increase without limit, we have
r (z) = ^ — /"(- yY^'e-ydy,
^ ^ 2 sin irzj ^ ^' ^
where the integral is taken along a curve commencing at positive infinity,
circulating round the origin in the counter-clockwise direction, and returning
to positive infinity again ; and in the integrand we must take (— yy^^ as
equivalent to e}*^^ ^^ ^■^^ where the real value of log (— y) is to be taken when
y is negative, and the logarithm is rendered one- valued by the stipulation
that the variable is not to cross the real axis at any point on the positive side
of the origin.
Since
r(5)r(l--^) = -r
IT
SmTT^
this result can be written in the form
77)-2^/(-y>"'^"^^y-
V{z)
This theorem is valid for all values of z — in contrast to that found in the
next article, which is true only for restricted values of the variable.
Example 1. Bourgvs^s expressions for the Oamma-function,
By a slight extension of the above proof, it is seen that
101] THE GAMMA'FUNCTION. 183
where the path of integration is restricted only to contain the origin and to be extended
indefinitely at both ends in the direction of the negative part of the real axis ; the
contour need not be closed.
Take then as contoui* two lines inclined at an angle a to the axis of Xy passing through
the origin, and a small circle round the origin. The integral round the small circle is zero
when z has its real part comprised between and 1. The integration along the two
lines gives the result
^^'^^-^ rp*-V«-*sin(p8in«+^)c;p,
which can be written in the form
Tiz)^-. \-. rp'-^ef'^^'' Bin (p-hza) dp,
^ ' sm zir (sm a)* J
This formula is true for all values of a which are not less than -^ . Taking a equal to
ir, we have the result
r(«)=| p'-^e-Pdp,
EaximpU 2. By taking for contour of integration a parabola with the origin as focus,
shew that
r(*)" 2,-ig-^^^ / «"**'(l+^*"*<5<>8[(2«-l)tan-ia?+^]cir. (Bourguet)
101. Expression of T {z) as a definite integral, whose path of integration
is real.
We have, by the result of the preceding article.
r (z) = =r^ { (5-V+ <«-« i<« <-y' dy.
Take a path ABODE, commencing at the positive infinitely distant
extremity of the real axis (which considered as initial point we denote by A\
proceeding close to the real axis until it arrives at the neighbourhood of the
origin, describing a small circle BCD round the origin, and returning, close
to the real axis, to positive inBuity again (which, considered as terminal
point, we denote by E). With the conventions that have been made, the
integral along the part AB of the path becomes
i f^
I e^+(*-i)logy-t>(z-i)^y
2 8m7r^;^ ^
in which log y is supposed to have its real determination.
The part of the integral due to the small circle BCD is easily seen to be
zero if the real part of z is positive. For the part of the integral due to DC,
we have
2 8in7rjrJo
184 TRANSCENDENTAL FUNCTIONS. [CHAP. IX.
Thus
28m TT^r Jo
or
Jo
This integral is called the Eulerian Integral of the Second Kind, It is
frequently given as the definition of the Gamma-function : but for this
purpose it is unsuited, since the integral exists only when the real part of g
is positive.
Example 1. Prove that when z is positive
Example 2. Prove that
Example 3. Prove that
* e'^x^'^dx
^X^'^dx^^\
(^+1)- • (^+2)* ' («+3)* • r(«);o «*-! '
102. Extension of the definite-integral expression to the case in which the
argument of the Gamma-function is negative.
The formula of the last article is no longer applicable when the argument
z is negative. Saalschiitz has shewn however that, for negative arguments,
an analogous theorem exists. This can be obtained in the following way.
Consider the function
r, (^)= JV> (^--1 +^- ^,+ ... + (- iy^'^)dx,
where ^ is a negative number lying between the negative integers — k and
-(A: + l).
By partial integration we have, when ^ < — 1,
r.w-[f («-■-.+.- ?,+...+(-!)"■ i;)]
09^ 00
102, 103] THE GAMMA-FUNCTION. 185
The terms in the left-hand member which are not under the integral sign
vanish, since (z + k) is negative and (-er + A + 1) is positive : so we have
The same proof applies when z lies between and — 1, and leads to the
result
T(z+l)=^zr,{z) {0>z>-l).
The last equation shews that, between the values and — 1 of ^,
r,(z)-r(5).
The preceding equation then shews that Fi (z) is the same as F (z) for all
negative values of z less than — 1. Thus for all negative values of z, we have
SaalschUtz's result
F(^)=jV^(6-«-l+a:-|^j.h.:.+(-l)*+^|^)^^
where k is the integer next less than — z.
Example. If a fimction P (ji)he such that for poeitiye values of fi we have
and if for negative values of fi we define P^ (ja) by the equation
where k is the integer next less than -^, shew that
1 1 .,-.., 1
AW=PW-- + fi-(;^)-... +(-!)*- ^7(;^
(Saalschiitz.)
103. 0atL88* eaypression of the logarithmic derivate of the Gammorfunction
as a definite integral.
We shall next express the function -7- logF(r) as a definite integral, where
z is supposed to be a positive real quantity.
1 r*
We have - = / er^dx,
8 Jo
Therefore loe « = I -de— 1 dx.
^ Jis Jo on
186 TRANSCENDENTAL FUNCTIONS. [CHAP. IX.
Thus we have
I e"*fi^"* log sds « I er's^^ds j
— dx
Jo ^ K Jo Jo )
r<.).rw/;^|<r--,ji^}.
This equation is due to Dirichlet.
Writing 1 -♦- a? = e* in the second term of the integral, and a? = ^ in the
first term, we have
which is Qauss' expression of -r- log r(^) as a definite integral.
^l0gr(«)- / I— Irr-^J rf^ . (Gauss.)
Example 1. Prove that
dz
Example 2. Prove that
d. ^,, , 1 P . r«(l-a) a(l-a)(2-a)^ "1
^■-/•■{q--}
104. Binet*8 expression of log T (z) in terms of a definite integral.
Binet* has given an expression for log r(^), which is of great importance
as shewing the way in which log F (z) increases as z becomes very large ; his
result will be used later in the derivation of the asymptotic expansion
of r {z).
We have by the last article (z being supposed real and positive)
r
changing -^ to ^ + 1, we have
5"«I'(' + ')-/"(t-*^)*-
Now f er^dt^^,
Jo ^
* JawmaX de Vie. Polyt xvi. (1S39), pp. 128—343.
104]
and
THE OAMUA-FUNCnON.
187
/,
— (e «
t
Jo Jo
Jo y
= log z.
Therefore
|iogr(.-Hi)=^+iog.+/;d*{-?^^-?:^
-< — /?-t«
,-e
-te ^
e
■^ t 6«-l
or
Integrate for z between the limits 1 and z ; so
iogr(z+i).|iog^+^(iog^-i)+j"J{^-l+l
i<«r(.)-(.-l)iog.-.+/;{j^,-l+^}r?
t
dt
dt
*'-fA^A*-^r'' ('>
The first of these integrals can be otherwise expressed in the following
way.
We have*
-l"^2 t Jo e^'^-1 '
Multipljong both sides of this equation by e~^ dt and integrating with
respect to t from zero to infinity, we have
^Jo (u' +
udu
(w*+i>*)(e^»-l)*
Integrating this equation from j> » ^ to /> = oo , we have
*•
i:?*(i^i4-i)-
tan"
■ ©
dt
6^-1
(2).
* A proof of this equation can be foand by making k infinite in the equation
2 -rs-i — 0=2 S I «-*•»« sin (ttt) dt*.
188
TRANSCENDENTAL FUNCnONS.
Thus equation (1) becomes
logr{z)^(^z~^logZ'Z'^2
tan""
■©"•
e«'»-l
^^ Jo V-l t^2
--dt
t
Now write z^^^ equation (1): since
r©
"»,
we obtain
1 , 1 r* ( 1 11
6
-i<
t
dt
-rA^.-w^T^
t
Write ^ for t in the last integral. Thus
or
=
=
g log IT
5 log IT
_i_r{_i 1 r
2 Jo V-1 ^^^I'^t
1 f» f -1 e-f"
2 Jo te'-l''" « ,
dt,
e-^) dt
t
Adding this to equation (3), we obtain
' 1\
logr(«) = lir-2Jlog«-« + 2
[chap. IX.
tan~M-)du , ,
.(3).
+Jo « v-i t (^-i^ t 2} ^*^-
The last integral is
rdt(l , tr*
-e-»«l
t
or
/ft J ^
JO
t
or
I log X dx,
* This ftrtifioe it dae to Pringsheim, MatK Ann, xxxi.
105] THE GAMMA-FUNCTION. 18^
or . 'l^ogl-^l.
Substituting this in equation (4), we obtain
logr(^) = ^^-2Jlog^-2: + 2log(27r)
g2iru_l
This is Binet's formula for log r(z)] ss z increases indefinitely, the last
integral diminishes indefinitely, and so the remaining terms furnish an
approximate expression for logr(^:) when z is large.
Example, Prove that
log r(«)=(«-i) log«-«+i log (2ir)+ J(«),
where J (z) is given by the absolutely convergent series
•^ ('^"* |m^ l"*" 2 (2r+l)V+2)'*' 3 (^+l)(« + 2)(«+3) ■*"•••/'
in which
^i~J> ^2~J> ^"fo> ^4"*Tcr>
and generally
c^z^j (x+l)(a?+2)...(a?+n-l)(2x-l)a7ctr. (Binet)
106. The Eulerian Integral of the First Kind,
The name Eulerian Integral of the First Kiixd was given by Binet to the-
integral
jB(p, ?)= [ x^^ {I - x)^-^ dx,
Jo
which was first studied by Euler and Legendre. In this integral, the real
parts of p and q are supposed to be positive ; and x^~^, (1 — x)^^ are to be
understood to mean those values of e^^^^**^* and e<^*^**^^""*^ which correspond
to the real determinations of the logarithms.
With these stipulations, it is easily seen that the integral exists, since the
infinity of the integrand is of less than the first order at the two extremities,
of the path of integration.
We have, on writing (1 - a?) for x,
B{p.q)=B{q,p).
Also
or
It
190 TRANSCENDENTAL FUNCTIONS. [CHAP. IX.
Also
Jo Jo
= Bip+l,q) + B{p,q+l).
Combining these results we obtain the formula
Example 1. Prove that if n is a positive integer,
r>/ i-iv * 1.2... tl
^(^'''+l)-p(^+l)...(p+n)-
Example 2. Prove that
106. Expression of the Evlerian Integral of the first kind in terms of
the Oamma-function,
We shall now establish the important theorem
^. ^ r(m)r(n)
B (m, n) = 't^^ — -/ .
^ * ^ r (m + n)
To prove this, we have
/•OO /•OO
r (m) r (n) = I e-%^*-^da; x | e-yy«-^ dy
JO ./o
(writing a^ for a:, and y" for y)
Jo Jo
4 / e-^'^-^^^af^^y^-^dxdy
J J
(writing r cos ^ for x, and r sin for y)
= 4 I f e-*«r*("*+n)-i cos*^^ sin«^-^ ^ dr dtf
^0 Jo
= r (m + n) 2 I cos«^» ^ sin*^i ^ d^
(putting cos* 0^u) = r (m + n) -B (m, n).
This result connects the Eulerian Integral of the first kind with the
Gamma-function.
106 — 108] THE QAMMA-FUNCnON. 191
Example. Prove that
(Cambridge Mathematical Tripos, Part I., 1894.)
107. Evahiation of trigonometrical integrals in terms of the Oamntar-
function.
We can now evaluate the integral
w
•a
I cos*'*~'a?8in**"^a:da?,
Jo
where m and n are not restricted to be integral, but have their real parts
positive.
For writing sin* a?= ^, we have
r* 1 P ^-1 5.1
1 r/^ n\
- 2 , (^.)
The well-known elementary formulae for the case in which m and n are
integers can be at once derived from this
Example. Prove that when | iE: | < 1,
pcoS^8in*^flW V 2 y \ 2 y P cos*»-"*^(f ^
jo O^siE^^jT" ^- /m+n+l\ -/<> (^.^^i^,^)!^-
(Trinity College Examination, 1898.)
108. DirichleSs multiple integrals.
We shall now shew how the integral
may be reduced to a simple integral, where / is an arbitrary function of
its argument, and the integration is extended over all the systems of
positive values of the variables x, y, z, which satisfy the inequality
192
TRANSCENDENTAL FUNCTIONS.
[chap. Et.
Write
111
a?«(L»i*, y^hyf, z^cz{'.
Then the integral takes the form
/ =
a^b^C
a/87
- jjJA^ + yi + ^) x^^^-^yi^^-^z^^-^dx^dy^dzu
where the integration is now taken over all the systems of positive values of
the variables ah, yi, ^1, which satisfy the inequality
^ + yi+'«i^i.
Now let
/i = ^+yi + ^i-f=0,
be three equations defining new variables f, 1;, f.
-1
a(/i,/2,/s)
Then
9(^1, yi, ^1)
-^
-1;?
1
1
1
1
1
1
=fV
The field of integration is clearly such that the new variables f , 1;, f, each
vary from to 1.
Thus
/ = ^^^^^ C f ^ f V(f ) P'+^»+''»-^ (1 - ^)^'-»i7«i-^'-^ (1 - O^^-'^-'d^dvd^
otp7 JoJoJo
a^7
-50),, 9. + r,)5(g.. r,)f /(f)P-^''+"-'df
Jo
"a^7
\a i8 7>
The multiple integral is reduced to a simple integral.
It is easily seen that this method of evaluation can be applied to multiple
integrals of a similar form in any number of variables.
109] THE GAMMA-FUNCTION. 193
Example 1. Shew that the moment of inertia of a homogeneous ellipsoid of unit density,
taken about the axis of «, is
-i(a«+i^«)fra5c,
where a, 6, c are the semi-axes.
Example 2. Shew that the area of the epicycloid ^ -)-y^s= ^ is |ir^.
Note, Dirichlet's integrals can also be evaluated by performing on the variables the
substitution
a?i«sf^ sin* $1 sin* ^„
yi «= r* sin* ^j cos* ^j,
«i»!r*cos*^i,
leading to the same result as above ; in the case of an integral with n variables the
corresponding substitution would be
a?i «r* sin* $1 sin* 6^ ••• sin* ^»- j, etc.
109. The asymptotic expansion of the logarithm of the Oamma-function
{Stirling's series).
We now proceed to 6nd an expansion which asymptotically (§ 88)
represents the function logr(i^), and is actually used in the calculation of
the Qamma-function.
For simplicity, we shall consider only real positive values of the argument
z. For a proof and discussion of the expansion when z has complex values
the student is referred to a memoir by Stieltjes*.
From Binet's expression for log F (z) (§ 104), we have
logr(^) = (^-5)log^-^ + |log(27r) + <^(^),
tan"' - ax
where ^(^) = 2/ ^^^
r) = 2J
* A
Now tan-- = ----+---...
(-1 y»-> ^-^ (-1) ^ r* t^dt
'*"(2n-l)~^^-i"^ z^^^ Jo t^ + z*
Substituting this in the integral, and remembering that
a?**"^ dx Br.
/
e»»*-l 4n'
* Lumvt2Ze*< J<mmal (4), v. pp. 425—444 (1889).
W. A. 13
194 TRANSCENDENTAL FUNCTIONS. [CHAP. IX.
where jBj, fij, ... are the BernouUian numbers, we have
Let us now find approximately the magnitude of the last term when z is
very large.
r* dx (^ t^dt
The quantity j^__J^__^
is less than
'''' ^1n+l)^«Jo
or
4(n4-l)(2n+l)z«*
If now any value of n be taken, it is clear that this quantity can be
made as small as we please by taking z sufficiently large.
It ollows that the quantity
|<PV^; r^i 2r(2r-l)^»-iJ
can be made as small as we please by taking a sufficiently large value
for z ; and therefore (§ 88) the series
-r
l,2.z 3.4.^ ■ 5,Q.z'
is the asymptotic expansion of the function <f> (z) for large real positive
values of z.
We see therefore that the series
l\i If /« N 5 (-ly-^Br 1
ar— 1
is the asymptotic expansion of the function logr(z) for large real positive
values of z. This is generally known as Stirling's series,
110. Asymptotic expansion of the Oamma-function,
Forming the exponentials of both members of the equation just found,
we have
S B S
r (z) = e-^z'-i(2v)i^''~^^^^^'"" ,
110] THE GAMMA-FUNCTION. 195
or
r (^) = «--^-i (2,r)* |l + ^ + § + . . .1 ,
where Cj = z-— ^ , C, = -^ , etc.
Substituting the numerical values of the Bemoullian numbers, the
formula becomes
n,z)-e z- \''^}'y-+i2z^2{12zy 30(12^)' 120(12^)* "'j'
This is the asymptotic expansion of the Oammorfunction, In conjunction
with the formula r(l -{-z)^ zT{z), it is very useful for the purpose of com-
puting the numerical value of the function.
Tables of the function log r {z\ correct to 12 decimal places, for values of t between
1 and 2, were constructed in this way by Legendre, and published in his Exercices de
Calcul Integral, Tome il p. 85, in 1817.
Miscellaneous Examples.
1. Shew that
<-.>(-l)('-3)(-l)--7(:;^-
(Trinity College Examination, 1897.)
2. If o-n be the sum of the n first terms of a divergent series
- + - + - + ...,
«i Oa «8
shew that the series
1 1 1
Ojo-i OfO-f a^^
is divergent.
If the squares of the terms of the latter series form a convergent series, shew that a
function O (l+z) can be defined by the equation
6^(1+^)= Limit
<rn'
and shew that
tf(T+7)-^°.{(i+i-J «■'••'-}>
where e is a constant. (Ceearo.)
13—2
196 TRANSCENDENTAL FUNCTIONS. [CHAP. IX.
3. Prove that
dz Jo !-«-«
-f"{(i+«)-'-a+«)-}T-r.
Jo fl
where y is Eider's constant.
4 Prove that
5. Prove that
r («) = Limit n'B («, n).
6. Prove that, when ^ > 1,
^(p, j)+^(p+l, j)+^(p+2, ?)+...-5(p, 9-1).
7. Prove that
Biscay q) gjr a(a + l)9(g + l)
^(i>i?) '*"l>+?"^1.2.(p + j)(p+j + l)"^--
8. Prove that
^tP, q)B(p+q, r)^B(q, r) B(q + r, p). (Euler.)
9. Prove that
log r (*)=(!-«) log ir+y(i-«)-i log sin «irH — 2 -^ — sin2n«9r.
IT n-i n
(Kummer.)
10. Prove that
I co8'>+«"*wco8(p-g)w(iM—; — ; Tv-r^rx:; ,-„-, r. (Cauchv.)
11. Prove that
..»i.(«,)-w(^)*/;'-!;=!^az|L'^.. m^->
12. Prove that
2»(p.i'+«) 2J— V+2(2^1) + 2".M2p+iK2^3) + "r ^ ^^
13. Prove that
r
r
14. Prove that
(j»+i)"lP 4p0>+l)^2.4».^0j+l)(p+2)^-; •
\ V(p) ] 2 t 2(2p+l) 2.4.(2p+3)(2p+6)^-r
MISC. EXS.] THE GAMMA-FUNCTION. 197
16. Prove that
(Binet.)
16. Prove that
where y is Eider's constant. (Legendre.)
17. Prove that
B(p,p)B{p+i,p+i)^,^^. (Binet)
18. If
shew that
/ \ogT{z)dz^u,
du ,
and hence (or otherwise) that
ti^orlog J?— ^+^log29r. (Raabe.)
19. Prove that, for all values of t except negative real values,
logr«-(f-i)log^-«+ilog(2^)+ i rTTl-
Sin %nirx
(Bourguet.)
n^r
20. Prove that
,_ r(a-hl)r(a+6+c4-l) Pfia-f^KLif!)^^
*'^r(a+6+l)r(a+c+l)"jo~;^ ' l ^-
(l-^)log-
21. Prove that
ri^«-i-^-i ^ V 2 / \2/
Jo(l-H^)loga?''^^ /a\ //3-Hy
(Kummer.)
22. Prove that
, r(a+6+l)r(a-Hc+l)r(6H-c-H) f Ml -^)a-^)(l-^) ^_
'^r(a+l)r(6+l)r(c+l)r(a+6i-c+l)";o ,, M 1
(l-x)log-
23. When a? is positive, shew that
- T{x) • 2nl 1
v;;-^w 2
(Cambridge Mathematical Tripos, Part I, 1897.)
24. If a is positive, shew that
r(g)r(a+l) ^ « (~ l)''a (g- 1) (a- 2) ... {a-n) 1
r(«+a) ^ »=o n\ « + »*
198 TRANSCENDENTAL FUNCTIONS. [CHAP. IX.
25. Shew that
26. The curve f^ «= 2** ~^ cosing is composed of m equal closed loops. Shew that the
length of the arc of half of one of the loops is
1 f' -i-1
2^"^.—. / (cosarV dx,
and hence that the total perimeter of the curve is
27. Prove that
logr(«)=(«-i)log«-«+ilog(2fl-)
28. Prove that
29. Prove that
30. Prove that
a- a*
log r («+a)=log r («)+a log « -
2«
a / a(l — a)<ia- / a{\^a)da
Jo Jo
2s (2+1)
a/ a(l-a)(2-a)flfot- / a(l-a)(2-a)tia
3«(«+l)(«+2)
31. Prove that
32. Prove that
5(p.?)-(^r,!.i(2-)*»*(p.«,
MISC. EXS.]
where
THE OAMMA-FUNCTION.
199
M(p,q)^2pj*^
dt
e^i^-l
(p+?)r
and
p*— ^+^+/>S'.
33. Expand
{r(a)}-i
as a series of ascending powers of a.
(Various evaluations of the coefficients in this expansion have heen given by Bourguet,
Bull, des Sci. Math, v. (1881), p. 43; Boiu^uet, Acta Math, ii. (1883), p. 261 ; Schlttmilch,
Zeit$ohrift fiir Math, xxv. (1880), pp. 35, 851.)
34. Shew that
and
where
r^^-a«oos 6a?<iir-co8 {(m + 1) («■ - <^)} ?^-^^^^
r^^—xsin &p<iir-sin {(m+ 1) (ir - 4)) ^^j- ,
- a-k-bi^r (cos 0+t sin ff>).
35. If
I e-'tf^idty
shew that
and
Pi^)--]
^^^^ X l!a?+l'*"2la?+2 3!^+3"*"*"'
P(x+l)-arP(ar)-i.
36. Prove that
d i-, r(z+ar) X ,x {x-l) . x{x-'\){x-'%)
5^*"* r(z) "« ^«(«+i)"*"^ «(«+i)(«+2) "^•••*
37. If a is negative, and if
where v is an integer and a is positive, shew that
r^r) r (a)
r(
where
(-l)»(a-l)(a-2)...(a-n)^
0.(,).gW-0(-«),
(Hermite.)
200 TRANSCENDENTAL FUNCTION& [CHAP. IX.
38. When - oo <a< 1, shew that
r{x)T(a-x) % K ; R*
r(a) ",!, ^+;i-,*,^::^:rn'
^^ (-l)»*a(aH-l)... (aH-n-1)
39. When a > 1, and y and a are reepectively the integral and fractional parts of
a, shew that
T{x)T{a-x) ^ J G{x)p^ • Q{x)p^^^
L^— a JP— a— I 4?— a — v+lj
where
*«-('-!)('-^)-('-,-T^)
and
_( - l)**o(a+l) ...(a+n-l)
40. If Pi, p^y ... py are the roots of the equation
p»' + aip''-i+...+aK«0,
shew that
• f/ X X* ' ^^ \ *!*)
,r(l)a.Xj^^^^
T (z- pix)r (z- p^) ...r {z- p^)'
41. If a and b are real and positive, prove that
/ e"i'V-iw«-*-irfttrfv»r(a)r(6).
42. By taking as contour of integration a parabola with its vertex at the origin, derive
from the formula
^W = s^ l^^^^dz
^ ' 2tsma9ry
the result
T{a)'^^^^f e-'^x^'i (I +j^)-^ [3 sin {x+acot'^i-x)}
+8in {x+(a - 2) cot-» {'-x)}]dx.
(Bourguet.)
43. Prove that, when 1 < « < 2,
/*sin6^ - __ ^"* «•
and when < « < 1,
sm-
pcosftg? . _ f-^
Jo A- 2r(«;
«■
) tr«'
MISC. EXS.] THE OAMMA-FUNCTION. 201
44. Shew that
w
46. If
2«
t7»
('-!)
and
s
2«
F-r
and if a function F(x) be defined "by the equation
shew (1) that F(x) satisfies the equation
(2) that for all positive integral values of x,
F(x)~T{x\
(3) that F{x) is regular for all finite values of x,
(4) that
46. Prove that the function O (x), defined by the equation
6f(a?+l)=(2ir)«e « « n fl+lj e» ,
has the properties expressed by the equations
log ^j^- , — (a» I «-^cotir:J?c2r-a?log2tr,
[ (g-l)(a;-2 ) «-l r Q +itn
(n+i)-i- (r(«+i)}.- in iij±;ij.
(Alexerewsky.)
47. If « is a positive quantity (not necessarily integral), and « is a real quantity
between - ^ and ^ , shew that
COS*
1 r(«+i) f. « „ . «(#-2) . . \
and draw graphs of the series and of the function cos's.
202 TRANSCENDENTAL FUNCTTIONS. [CHAP. IX.
48. Obtain the expansion
and find the values of x for which it is applicable.
49. Prove that
(Cauchy.)
i.(-5)' '^^"
60. If
('•*^-rV)/o
where | ^ | < I and the real part of or is positive, shew that
and
Limit (1 - x)^-* f («, ^) = r (1 - *).
ae-l
61. If Xf Wy and « be real, and < t9 < 1, and « > 1, and if
shew that
and
00 ^iri*
*(«.,;r,l-,).^J ^'' I. (Lerch.)
62. If
shew that
tf'-l '
(3) r(|) n-iCW-rQ^^ ,r*Tf(l-,).
63. Let the function <f>i''>(x) be defined by the equation
(-!)•*«(
I n(i-«— ^)
where 5 is an integer ^ m, the function x (0 ^^ defined by the equation
and the quantities a^ are constants whose real part is positive.
MISC. EX&] THE QABfMA-FUNCTION. 203
Shew that ^t*) (x) can be expressed by the series
<^«(^)-2/W(jp+M^);
where t9«>ZXrart
and where
(-i)'/«(«)-j'"x(0 «•«-"*•
Shew also that ^t*) (x) satisfies the functional equation
M.»'
Shew further that when ;^(r)«Bl, <f>i')(x) becomes a function ^^(x\ which has the
multiplication-theorem
where all the quantities X vary from to (n- 1).
(Pincherle.)
64. If
where
/n\ n!
W"rl (n-r)!'
shew that
. riy)r(sf'-x+n)T(x+v)T{v+n)
/nwy, «^; r(y-i?)r(y+n)r(t^)r(a?+v+n)'
and that
r(y)r(a?+v)
(y-x-l)r(-n)r(a?+l)r(y+t;+n-l)'
(Saalschiits.)
CHAPTER X.
Lbgendbb Functions.
111. Definition of Legendre polynomials.
The expression (1 — 2zh + /i*)"^
can, when | A { is sufficiently small, be expanded by the multinomial theorem
as a series of ascending powers of h, in the form
where Pi {z) = Zy
-P« (^) = ^^-^— ^ . etc.
The expressions Pi{z), P,(^) ..., which are clearly all polynomials in z,
are known as Legendre polynomials, Pn (z) is called the Legendre polynomial
of order n.
It will appear later (§ 116) that these polynomials are particular oases of a more
extended class of functions, known as Legendre functions.
Example 1. Prove that
P„(coe^)=^ — r^coseC*-^^^ ,/ ^'
" ^ n ! a (cot B)^
(Cambridge Mathematical Tripos, Part II, 1893.)
Let ^ be an angle such that
Then
(1-
-2A
cos^+A«)"
4_8in ff
"~ sin ^ *
cot^« -r-=-^-
*
coe^-^
~ sin^
■scot^-A
cosec
B.
Ill, 112] LBGENDBE FUNCTIONa 205
Therefore by Taylor's theorem we have
• iK « ( - A cosec ^)» €?* (sin ^)
sin ff ^* ;; « V . ^x" •
n n\ (f(oot^)»'
or
/, or ii.nv-* ^(-A)*coseo»+»^flP«(8m^)
(l-2Acoe^+A») *=S^-^ir! d^r
Equating coefficients of A*, we obtain the required result.
Example 2. Shew that
For
/>-©'■
Therefore
Thus
(l-2A*+A«)-*=ir-* r «-(i-«'+*^««cft=ir-* f * tf-(i-'«)««6-(»-*y««(&.
y —00 J —00
whence the result follows.
Example 3. By equating coefficients of powers of A in the expansion
(l-2Aco8^+A*)* \ * 2.4 /
X (n-iAe-«+^A«d-««+...),
shew that
112. SchldfiVs integral for Pn{z\
Let h be any quantity which is not greater than the radius of convergence
00
of the series 2 h^Pn {z\
Then (1 — 2zh + A*)~* can be expanded as the series
1+AP^(^) + A'P«(^)+A»P3(^)+....
But (1 - 2zh + A')~* is the residue, at the pole
1 (1 - 2zh + /i')*
^"h" h
of the function — 2A""* "^ I ^ "" t J Ti r •
206 TEANSCENDENTAL FUNCTIONS. [CHAP. X.
Now the last expression has two poles, namely at the points
1 (1 - 2zh + h*^
and t^Ud^^^^,
ft n
When h is very small, the former of these poles is close to the point
t = z, while the second pole is in the infinitely distant part of the plane.
Therefore, if (7 be a contour in the ^-plane, including the point z, the former
pole only is contained within C when h is not large, and so we have
di.
Equating coefficients of A", we have the result
which is called Schldflis integral-formula for the Legendre polynomials
113. Rodrigues* formula for the Legendre polynomials.
From Schlafli's integral
dt
we immediately deduce, by the theorem of § 38, the result
which is called Rodrigvss formula,
114. Legendre'a differential equxttion.
We shall now prove that the function y = Pn (-^) is a solution of the
differential equation
which is called Legendre's differential equation of order n.
* Schlafli, Ueber die beiden Heine*$cheii Kugelfunctionen ; Bern, ISSl.
113—116]
LEOENDRE FUNCTIONS.
207
For on substituting Schlafli's integral, we have
ds^
dz
27n
and this integral is zero, since the function (<* — 1)**"*"^ {t — ^)~"~' resumes its
original value after describing the contour (7. The Legendre pol3n[iomial
therefore satisfies the differential equation.
The differential equation can clearly be written in the alternative form
116. The integral-properties of the Legendre polynomials.
We shall now shew that
and that
j[Pm{z)Pn{z)dz = 0,
if m and n are positive integers and m is not equal to n.
dPn]
For since
we have
+ (m-n)(m + n-l)\ PnPndz^O.
■ —1
Integrating by parts, this equation gives
- (m - n) (m + n - !)/'-?». («) -P» («) dz
-['a-^){p„(.)^-P„w^)};
which shews that the integral
j' P„(z)Pn{z)dz
has the value zero when m is not equal to n.
= 0,
208 TRANSCENDENTAL FUNCTIONS. [CHAP. X.
To establish the second part of the theorem, let the equation
be squared, and the resulting equation integrated between the limits — 1 and
+ 1 ; using the result already proved in the first part of the theorem, we thus
obtain
or
Equating coeflScients of h^ in this equality, we have
which is the desired result.
Example 1. Prove that, if m is not equal to n,
x{l +(-!)*+«}.
(Cambridge Mathematical Tripoe, Part I, 1897.)
Example 2. Prove that
j-i dzr d^ ^^ '^ '^^'''^ (27i + l)(n-r)!'
according as m, n are unequal or equal.
(Cambridge Mathematical Tripos, Part I, 1893.)
116. Legendre functions.
Hitherto we have supposed that the index n of Pn{z) is a positive
integer; in fact, Pn{z) has not been defined except when n is a positive
integer. We shall now see how the definition can be extended so as to
furnish a definition of Pn{^)» even when n is not integral.
An analogy can be drawn from the theory of the Gamma-function. The expression
zl as ordinarily defined (viz. as « («— 1) (*-2)...2. 1) has a meaning only for positive
integral values of z; but when the Gamma-function has been introduced, z\ can be defined
to be r (2 + 1)) and so a function z\ will exist for all values of z.
Referring to § 114, we see the differential equation
116] LEQENDRS FUNCTIONS. 209
is satisfied by the expression •
even when n is not a positive integer, provided that (7 is a contour such that
the function
(^ - l)n-fi
{t - xr)»+a
resumes its original value after describing C.
Suppose then that n is no longer taken to be a positive integer.
Now the function (f*— 1)*+* (t — xr)-**-« has three singularities, namely the
points t^l,t^^l,t»z; and it is clear that after describing a small closed
contour enclosing the point ^^l, the function resumes its original value
multiplied by e»»»<»+i) ; while after describing a small closed contour enclosing
the point t = z, the function resumes its original value multiplied by
If therefore (7 be a simple contour enclosing the points t^ I and t ^ z,
but not enclosing the point ^=: — 1, then the function
(e» - l)n+i
Ji+a
(t-z)
will after describing C resume its original value multiplied by e"***, i.e. it will
resume its original value. Hence whatever n he, Hie Legendrian differential
equation of order w,
is satisfied hy the expression
where is a simple contour in the t-plane enclosing the points t^l and t^g,
hut not enclosing the point ^ » — 1.
This expression will he denoted hy P» {z\ and will he termed the Legendre
function of the first kind and of order n.
We have thus obtained a definition of Pn {z) which is valid even when n
is not integral.
The Legendre function is a mere polynomial when n is integral, but is
a new transcendental function when w is not integral ; just as F {z) is the
polynomial (^-* 1)1 when z is integral, but is a transcendental function when
z is not integral.
W. A. 1*
210 TRANSCENDENTAL FUNCTIONS. [CHAP. X.
We shall suppose the many-valued function ^, which occurs in the
defining integral, to have the value 1 when z is equal to 1, and when z
is not equal to 1 to have that value which would be obtained by con-
tinuation along a rectilinear path from the point 1 to the point z,
117. The Recurrence-formulae,
We proceed to establish a group of formulae which connect Legendre
functions of different orders.
We have by § 116, for all real or complex values of w,
m
Integrating by parts, we have
27ri;e2»-(<-«)"
and hence we have
f „(.)-.P^. (^)- 2^/, 2^^(7-1;^. ^^ <^)-
Differentiating this equality, we obtain
—dz '~~d'z ■^"-' ^'^ -~2^lc S^'it-zr*^ - ^" ^^^»-' ^^^'
80
^>-.%i^>-nP„_.(.) (I).
This is the first of the required formulae.
Next, from the identity
we deduce
or
r (f'-l)»-> ^,, f 2 Kf - 1) + 1} (n - 1) (f - If-' dt
[ (n - 1) {(t -z) + z] (f - ly-' ^,
117] LEGENOEE FUNCTIONS. 211
or
or
(n-i)^r o*-iy'-^
27n; ./c2'*-H^-'8^V
or, by formula (A) above and Schlafli's formula,
= n{PnW--^Pn-iW}+(w-l)Pn-.(^)-(^-l)^Pn-iW,
or
nPnW-(2n-l)5Pn-i(^) + (n-l)Pn^(^) = (H),
a relation connecting three Legendre functions of consecutive orders. This
is the second of the required formulae.
Other formulae can be deduced from (I) and (II) in the following way :
Differentiating (II), we have
"" --^— -(2n^l)^ d^ "^^^"^^"d^ =(2n-l)P^,(4
Substituting for
dPn (Z)
dz
from (I), we have
f dPn-x^) p ,1 dPn-x(^)^,^ ,, dPn->(^)
= (2n-l)P„_,(^),
or
-(n-l)r^^g<l> + (»-l)^^g^^ = -(n-l)'P^,(4
or
^ —Tz -^^ =(n-l)P„-,(4
Changing (n — 1) to n in this equality, we have
Next, changing n to (n + 1) in (I), we have
14—2
r^^^BiB^BR
212 TRANSCBNDENTAL FUNCTIONS. [CHAP. X.
Adding this to (III), we have
^-P^^(^) - ^-P^^(^) ,(2n + l)Pn(^) (IV).
Lastly, combining (I) and (III), we obtain the result
{ii>-l)^^^nzPniz)-nP^r{z) (V).
The formulae (I) to (V) are called the recurrence'/ormtdde.
The above proof holds whether n is an integer or not, Le. it is applicable to the general
Legendre functions. Another proof which, however, only applies to the case when n is a
positive integer (i.e. is only applicable to the L^;endre polynomials) is as follows :
Write
Then equating coefficients of powers of A in the equality
(l-2Az+A«)|^=(«-A)r,
we have
nP,W-(2n-l)«P^_iW+(n-l)P,,.,W-0,
which is the formula (II).
Similarly, equating coefficients of powers of A in the equality
we have
which is the formula (III). The others can be deduced from these.
Eixtmple, Shew that, for all values of n,
(2n+3)P«,^i-(2n+l)P.«=^{z(P,«+P«,+i)-2P»A+J.
(Hargreaves.)
For
^{^W+i^n*i)-2P,P,^J
=P.«+P«..,+2.P.5^+2.P..,%^-2P.%.2P.,,f^
=P,«+i»,+,+2P,(-n-l)i>.+2P,^i(n+l)i'.+ ,
(as is seen by using formulae (I) and (III))
-(2»+3)i«,^,-(2n+l)P,«,
which is the required result
118] LEGENDRE FUNCTIONS. 218
118. EvaiuaUon of the integral-expression for Pn (z), as a power-series.
When n is a positive integer, we have seen that Pn {z) is a polynomial in
z. When n is not a positive integer, however, P^ {z) is not a polynomial ;
and as P^ (z) is not a regular function of z for all finite values of z unless n
is integral, it follows that no power-series exists which represents P^ (z) for
all finite values of z, when n is not integral. In order to find a power-series
capable of representing Pn(z), we must therefore make some supposition
regarding the part of the 5-plane on which the point z lies. We shall suppose
that z lies within a circle of radius 2, whose centre is the point 1 ; so that
|1-^|<2.
As the contour C of § 116 was subject ooly to the condition of enclosiug
^ a> 1 and t^z without enclosing t » - 1, it is clear that we can choose it so
as to lie entirely within the circle of centre 1 and radius 2 in the f-plane,
ie. to be such that the inequality 1 1 * f | < 2 is satisfied for all points t on C.
Now write ^— 1 = (^ — l)t^. When t describes the contour C, the point
representing the variable u will describe a contour 7 on the t£-plane ; since
C encloses the points t^z and ^ » 1, 7 will enclose the points u^l and u^O;
2
and since |1 — <| < 2, we shall have \u\< . ^ . for all points u on 7.
Then changing the variable of integration from t to u ih the integral
which represents Pn (z), we have
^(^•^V-T^'*^--^)-^^
2wi.y
2
Since \u\ < -, 77 we can expand this in the form
du.
Now on integrating by parts, we have the result
J y Ly W J Tt J y
The first expression on the right-hand side is zero, and so we have
I !*•'+*• (u-l)-*^»dw«^^^[ w«'+»-»(w-l)-~-*(u-l)dw
n Jy ^ ' n Jy
or
I w'+« (u - !)-*-> du = ^^i^ I w'"+«-^ {u - 1)-«-* du.
214 TRANSCENDENTAL FUNCTIONS. [CHAP. X.
Therefore
Now transform the integral on the right-hand side, by writing u ss
v-l*
r f t?*dt;
The integral I w*(tt — l)~*~*(iu becomes — I — -r , where the integration
has now to be taken in the positive sense round a contour 8 enclosing the
points t; » and t; «= oo , but not enclosing the point v^l. This integral can
be replaced by + i --— j , where the integration has to be taken in the
positive sense round a contour S^ enclosing the point t;= 1, but not enclosing
the points v^O, or t; ae oo (since the integrand has no singularities in the
region between the contours B and S^). The contour B' can now be diminished
until it becomes an infinitesimal circle surrounding the point t; = 1. The
value of the integral is then 1* I -— ^ , where the integration is taken round
this contour; or 2m, since the many- valued function v* has been taken
to have the meaning 1 at the point t; = 1. We thus have
r + n r — 1 + n 1 + n
2mjy ^ -/ -- ^ • ^.1 ••• 1 '
and on substituting this in the expression already found for Pn(^)i we
obtain
p , X 1 _^ S n(n-l)...(n-r-H) (r-hn)(r-l+n)...(l-f n) fz-lV
an expansion of Pn (z) as a series of powers of ('2^—1).
If now, as in § 14, a series of the form
^^l.c^^ 1.2c(c-hl) ^+-
(a hypergeometric series) be denoted by
F(a, 6, c, z\
then the expansion can be written in the form
Pn(r)-J'(-n, n + 1, 1, ^^).
This is the required expression for P^ (z) as an infinite series. It is valid
at all points z within the circle whose equation is |1 — ir| < 2.
119] LEGENBBE FUNCnON& 215
Corollary, Since this series is clearly unaffected when n is changed to
— n — 1, we have
Note. When n is a positive integer, the above series terminates and gives the expression
1 — «
of P» («) as a polynomial in —^ .
119. Laplace's integral-expressuyn for P^ {z).
We shall next shew that, for all values of n and for certain v^alues of z,
the Legendre function Pn{z) can be represented by the integral (called
Laplace's integral),
- ['{^ + cos <^ (^ - l^}«d<^.
When n is not an integer it is necessary to state which of the branches of
the many-valued function in the integrand is to be taken: we shall take
that branch of the function [z + cos <f){^ — 1)*}** which reduces to unity when
taken by the process of continuation along a straight path to the point z^l.
It will appear later that it is immaterial which branch of the two-valued
function (^ — 1)* is taken.
(A) Proof applicable only to ike Legendre polynomials.
When n is a positive integer, the result can easily be obtained in the
following way. We have
2 A«P«(^)-(1-2A^H-A«)-*.
But
(i-2A.+A«)-*=ir^^f-^^
d<f>
(l-A^)-A(-2»-l)*cos<^'
as is seen by applying the ordinary formula for the integration of
d<t>
:
a + 5cos<^*
Expanding the integrand of the integral in ascending powers of h, we have
(1 - 2A^ + A«)-* « - i A« ['{z + cos ^ (^«- 1)*}**^^,
^»=o Jo
and on equating coefficients of h^ on the two sides of this equation, the
required result is obtained.
As however the theorem is true whether n is an integer or not (ie. as it
is equally true for the Legendre functions and the Legendre polynomials),
it is necessary to have a general proof independent of the character of n ;
this will now be given.
(B) General proof
First, we shall shew that Laplace's integral satisfies Legendre's equation.
I^^p*^^^^^^"
216 TRANSCENDENTAL FUNCHONa [CHAP. X.
For if we write
'fJo
we have
= - ['{^H- COS <^(^«- !)*}«-» {n8in»^-l-^C08^(^-l)-*}d^.
But
f '{-? + COS <^ (z« - !)*}«-« sin«<^d^
« — {^ + cos ^ (j8* - 1)*}*^ 8in<^ COS ^
4- Tcos A :^ [sin <^ [-? + cos <^ (i^ -!)*}«-•] d^
« / ' {^ + cos ^ (^ - 1 )*} »-« COS" <^ rf^
- (n - 2) f '{^ + COS <^ (z« - 1)*}'^» cos <^ (-?» - 1)* sia«^d<^
-'o
«r{^ + cos^(^«-l)*}~-^ci<^~(n-l)r{xr + cos<^(-8«--l)*}»-^sin«<^d^
Jo Jo
+ (n - 2) ^ rsm*^dif> {2: + cos ^ (-e* - 1)*}**"*.
Jo
Therefore
n 1 {-ar + cos ^ (-«■ - 1)*}*^ sin«^(i<^
Jo
=rf'{2r + cos^(-e»-l)*}'«d<^ + (n--2)2rr{ir + cos<^(-^«-l)*}~-«sin«<^^^
Jo Jo .
Thus we have
(l-^)g-2.| + n(„ + l)y
= - (n - 2) « f ' {« + COB (^ («• - !)*}"-• sin'^d^
9r Jo
- - « (^ - 1)-* f'fz + cos ^(«' - l)»}»-'cos <tKi4>
= « !L ^ (^ _ 1)-* f ^ [U + cos ^ (^ - 1)*}"-* sin ^] #
TT Jo w<P
-0,
119] LEGENDRE FUNCTIONS. 217
which shews that Laplace's integral satisfies Legendre's equation, whatever
n and z may be.
1 -^z
Now suppose that z is nearly unity, and put — ^— » u. Then the integral
becomes
1 r*
-I {l-2w + cos^(-4w4-ti«)*J«d<^,
which for small values of u can be expanded in the form
1+- d<i 2 -^^ ^- — j^^ ^ {-2a + co8 A (-4w + w*)*K
This is a series of powers of m*; the first terms (neglecting w*) are
If* If'
1 + 2inw* - I cos ^d,^ — %nu - / {1 + (n — 1) cos'^} d^,
or 1 — inu ^ ,
2
or 1 — w (w 4- 1) «.
It is clear that odd powers of 'v^ can arise only in conjunction with odd
powers of cos^ in the integrand, and so here vanish when integrated.
Laplace's integral can therefore, when u is small, be expanded in ascending
powers of u in the form
1 — n (n + 1) w + OjW* + OjW* + a4W* + ... .
But the coefficients a,, a,, ... can be found by substituting this expression
in Legendre's equation, and equating to zero the coefficients of each power
of u. We thus find that
/ txy ^(^-~l)»"(^^y + 1)'(1 -l-n)...(r — 1 +n)(r-hn)
r I r :
and thus Laplace's integral is equal to
jP(-n, n + l, 1, -^)»
or (§118) to .PnW.
We thus have, for all real or complex values of n, the result
T^(z) « 1 ['{^ + cos <^ {z^ - 1)*}» d^.
It must be observed that as the power-series Fi-^n^ « + l, 1, — g— j
was used in the proof, this proof is valid only for values of z which satisfy
218 TRANSCENDENTAL PUNCTrONS. [CHAP. X.
1 1 — i^l
the inequality - — ^ — < !• ^^ however Pn (z) is an analytic function of z,
'the result will be true for a more extended region including this, provided
the integral
is an analytic function of z within this more extended region: since if
these two expressions are equal for any region however small in which they
are analytic functions, they must be always equal so long as they remain
analytic functions. But it is easily seen that for the integral
i J'{^ + cos <^ (^« - l)*j« d<^,
every point on the imaginary axis in the -^-plane is a singularity: and
therefore the region in the ^^-plane for which the equality
"Jo
is established is the region for which the real part of z is positive.
Corollary. Since
we have for all values of n, real or complex, the result
Pn (Z) = - ['[Z + COS <^ (^ - l)»}-^» d<l>,
TrjQ
so long as the real part of ^ is positive.
Example, If
shew that
(Binet)
120. Tlie Mehler-Dirichlet definite integral for P« (z).
Another expression for the Legendre function as a definite integral may
be obtained in the following way.
For all values of », we have by the preceding theorem
P„ (^) = 1 Tf^r + cos ^ (^ -!)*}« d<^
In this integral, replace the variable <^ by a new variable h, defined by
the equation
A «= ^ + (^ — 1)*C08 ij>
120] LEGENDRE FUNCTIONS. 219
80 that
and
We thus have
ctt«-(^-l)i8in<^d^,
i(l-2A«: + A«)*-:(^-l)*8in<^.
and therefore
(1-2A^ + A»)*
d<f>,
Now write z = cos 0. Thus
Pn(co8^)«- C^A'*(l-2Acos^ + A«)-*dA.
Writing A « «•♦, this becomes
^^^'^^^~^j.,(2cos<^-2cos^)*'
or
i> / ^\ 2 r^ cos (^ + i) i , ,
Pn (cos ^) » - I 7^7 ^T ^loi #.
^ ' ^ J {2 (cos <^ — cos 0)}^ ^
This is known as Mehlers simplified form of Dirichlet's integral. The
result is valid for all values of n.
Example 1. Prove that, when n is a positive integer,
*^ ' irj^ {2(cosd-co8<^)}*
For we have
Put
r* dw IT
Jo a+6-(a-6)co8ir*2a^*'
a-(H-A)«, 6=l-2Ay+A«,
The equation becomes
IT /-y (i±.^)f^
Writing £«cos <^, y>«coB ^, this gives
(l-2Acos^+A<)-*=- f'(l+A)8in<^(l-2Aco8<^+A2)(l+co8<^)-i(oo8^-co8<^)-*ci0.
ir J
Equating coefficients of A* on both sides, we have
P,(cos^=^| ^«in(n+^)<^sin<^
{2 (cos ^- cos 0)}*
^ ]. r* sin (nH
I sm^coS"^
./ A 2 2
220
TRilNSCENDENTAL FUNCTIONS.
[chap. X.
or
P»(co8^-- I {2 (006^-008 0)}-4 sin (n+i)<^.
Example 2. Prove that
PAoo.e).±.f^
2Aco8^+l)i
dh.
the integral being taken along a closed path which encloses the two points h^e*^^ and the
conventional meaning being assigned to the radical.
Hence (or otherwise) prove that, if ^ lie between }ir and }ir,
2.4... 2n
^-^^^^^-nzr^ik^)
coa(nB+<t>) V C06(n^-h3<^)
(2sintf)* ''■2(2«+3)~'(2sin^)l
1«.3* oos(n^+6<^)
where <t> denotes i^-^ir.
Shew also that the first few terms of the series give an approximate value of F^{cob$)
for all values of $ between and n which are not nearly equal to either or n. And explain
how this theorem may be used to approximate to the roots of the equation P^ (cos ^=0.
(Cambridge Mathematical Tripos, Part II, 1895.)
121. Expansion of Pn(z) as a aeries of powers of - ,
z
We now proceed to find an expansion of the Legendre function which is
valid for large values of z.
If the real part of 2^ be positive, we have for all values of n (fix>m
Laplace's integral)
^ Jo
Now suppose that |-?| is very large : then this can be written in the form
Expanding the integrand in ascending powers of - , this gives
z
We can evaluate
/ (1 + cos <^)'* d<^ and | cos<^(l + co8<^)'*-*(i</>
^0 Jo
121, 122] LEGENDBE FUNCTION& 221
by patting ^ » 2*^ and using the result
and thus we find that P» {t) can be expressed by a series of powers of - , the
first two terms of the expansion being given by the equation
P .V 2»5"r(n+^) ( _ n(n-l) )
The general law of the coefficients in the series can without difficulty be
found by substituting in Legendre's differential equation (§ 114) ; and iu this
way we find that P» (z) can be eaypressed by the hypergeometric series
P/^^ = 2VM[>^) /1-n n 1 1\
in the notation of § 14.
This series has only been proved to hold when z is large and the real part
of <g: is positive : but by § 14 it converges, and so represents an analytical
function, over all the area outside the circle of centre and radius 1. The
series therefore represents Pn(z) over this region.
122. The Legendre functions of the second kind.
Hitherto we have considered only one solution of the Legendre differential
equation, namely Pn (z). We can now proceed to find a second solution.
It appears from § 114, that the differential equation
is satisfied by the integral
^t* - l)"" (t - zy-' dt,
/<
taken round any contour such that the integrand resumes its initial value
after making the circuit of it. Let D be a figure-of-eight contour in the
^plane, enclosing the point t^ + 1 in one loop and the point t = — 1 in the
other, and not enclosing the point t^z. Then after describing this contour,
the above integrand clearly resumes its initial value, since it acquires the £su;tor
e^ after describing the first loop, and this is destroyed by the &ctor e"****
acquired during the description of the second loop. D is therefore a possible
contour.
A
222 TRANSCENDENTAL FUNCTIONS. [CHAP. X.
A solution of Legendre's equation is therefore furnished by the function
Qn {z)y if Qn {z) be defined by the equation
it is supposed that, in describing D, the point t makes a positive, i.e. counter-
clockwise, turn round the point ^ = — 1, and then a negative, i.e. clockwise,
turn round the point ^ » + 1. The significance of the many- valued functions
(^• — 1)" and {t^z)"^"^ will be supposed to be fixed in the same way as
before.
Another form of the integral may be obtained in the following way.
Let the contour become so attenuated as to consist simply of a line
joining the points — 1 and + 1, described twice, and two small circles round
the points — 1 and + 1 : when the real part of (n -h 1) is positive, the parts
of the integ^l arising from these two loops are at once seen to be infinitesi-
mal; and thus we have
= 2isin?i7rJ {l-fYit-zY^'^dt,
«o Qn{z)^^,\\\-i^T{z-t)^'di,
This last result is valid when the real part of (w + 1) is positive. When n
is a positive integer, the original definition of Qn{z) becomes imdeterminate:
in this case we can use the formulae just found.
Qn{z) is called the Legendre function of the second kind and of order n.
123. Expansion of Qn (z) as a power-series.
We now proceed to express the Legendre function of the second kind as
. . 1
a power-senes m - .
We have, when the real part of (71 -f- 1) is positive.
Suppose that 1 ^ 1 > 1. Then the integral can be expanded in the form
«.(«)-ss^ /la -«■)■ (1 -;)""' <»
123] LEO£NDBE FUNCTIONS. 223
as is seen on writing r for 28, since the integrals arising from odd values of r
obviously vanish.
Writing ^ « M, we can evaluate the coeflScients of powers of - as follows :
z
r(n + i)r_(* +i)
r(n + « + f) "'
and thus the formula for Qn {z) becomes
Q (^\J'^^ r(n4-l) 1 y/n-hl 71 + 2 3 1\
VnW 2»^ir(n + f)^«+i^ V 2 ' 2 ' ^^r z'J'
This is the expansion of the Legendre function of the second kind as a
power-series in - , corresponding to the expansion obtained for Pn (-?) in § 121.
z
The proof given above applies only when the real part of (n + 1) is positive ;
but a similar process can be applied to the integral
^ ^ 4i sin riTT j j[) 2^ ^ / \ /
the coefficients being evaluated in the same way as those which occurred in
the expansion of the Legendre function Pn (^) in ascending powers of —5— ;
the same result is reached, which shews that the formula
O r^^ '^^ r(n-H) 1 r^/n-H n + 2 3 1\
VfiW*2«+ir(n + f);?«+^^ V 2 ' 2 ' ^"^2' W
is true for all values of n, real or complex, and for all values of z represented
by points outside the circle of centre and radius unity.
Example 1. Shew that, when n is a positive integer,
We can write Legendre's differential equation in the form
(l-^g-2*|+n(«+l)«-0.
It is easily verified that this equation can be derived from the equation
(!-«») 5+2(»-l)«J+2«*=0,
by differentiating n times and writing t£» ;^ •
224 TRANSCENDENTAL FUNCTIONS. [CHAP. X*
Now one solution of the latter equation is a7«(2'- 1)* ; and a second solution oan be
derived bj the ordinary process for finding a second solution of a linear differential
equation of the second order, of which one solution is known. Thus two independent
solutions of this equation are found to be
It follows that
(««-l)» and (««-l)» I (t^-l)-«-irfr.
is a solution of L^;endre's equation. As this expression, when expanded in ascending
powers of - , commences with a term in d'^'^ it must be a constant multiple of Q^ (t) ; and
on comparing the coefficient of «"*~* in tnis expression with the coefficient of z"^"^ in the
expansion of Q^ (')» ^ found above, we obtain the required result.
Example 2. Shew that, when n is a positive integer, the Legendre function of the
second kind can be expressed bj the formula
For on expanding the int^;rand (v^-l)"'"'^ in ascending powers of -, the right-hand
V
side of the equation takes the form
^" ax- /.■'*>"■ {^' - SSI- '^^^m^ -..} .
and on performing the integrations this becomes
n! f 1 ■ (n-H)(7t+2) 1 1
, (2» + l)(2»-l)...3.1 |«* + i"*" 2(2»+3) z^+s -«■—/»
or §«(«).
Example 3. Shew that, when n is a positive integer.
This result can be obtained by applying the general integration-theorem
to the preceding result.
124. The recwrrence-formvlae for the Legendre function of the second
kind.
The functions PnW and Qn(^) bave been defined by integrals of pre-
cisely the same form, namely
/(t«-i)»(^-^)-«-id^.
It follows therefore that the general proof of the recurrence-formulae for
Pn(^)t given in § 117, is equally applicable to the function Qn(^)'i a^id hence
that the Legendre function of the second kind satisfies the recurrence-fbrmtdae
124, 125] LEGENDRE FUNCTIONS. 22&
dQn(z) dQ^Az)
^Z
g^' = n(2^.(4 ,
dz
nQn (z) - (2w - 1 ) zQ^, (z) + (n-l) Q^ (z) = 0,
, dQn (Z) dQn-r (Z) _ ^^ , .
dQn+1 JZ) dQn-i (Z) ,^_ . , , ^ .
-rfi d^^ (2n+l)Q„(.).
ft
(^ - 1) ^^ = nzQ„ (z) - riQ^, (z).
126. Laplace's integral for the Legendre function of the second kind.
Consider the expression
y = f {z + cosh (z* - 1)*}-**-^ d0,
Jo
in which z is supposed not to be a real negative number between — 1 and
— 00, and the real part of (n + l) is supposed to be positive; under these
conditions the integral certainly exists.
If now we form the quantity
(which occurs in Legendre's diflferential equation), we find for it the value
- (n + ly i {z-k- (f - 1)* cosh ^}-^« sinh* Odd
Jo
+ (w + 1) f {2: + («» - 1)* cosh e]-"*-* dd
Jo
4- (n + 1) 2r («« - 1)-* f {^ + (^ - 1)* cosh ^j-"-' cosh dd0.
Jo
This expression can be transformed, by integration by parts, in exactly the
same manner as the corresponding expression found in the discussion of
Laplace's integral for Pn (z), in § 1 19 ; and thus it is found to be zero. The
quantity y therefore satisfies Legendre's equation.
In order to compare y with the solutions Pn (z) and Q» (z) which have
already been found, we suppose that | ^ | is large, and write y in the form
g-nr-i
J*|l+cosh^(l-^ + ...)p 'd0,
W. A. 15
226 TRANSCENDENTAL FUNCTIONS. [CHAP. X.
which when expanded as a power-series becomes
^0 a^ a^ ch^ m
where ao= f (1 + cosh ^)-«-i d^
^0
/,
'^,.v'(l-v)-*dv. where . = j-j-^.
= ^— rr-8(^ + l> i)> where B is the Eulerian
2n+i
integral of the first kind,
Now any expression of the form (1) which satisfies Legendre's differential
equation must be a multiple of On(^) (since, by substituting the expansion in
the differential equation, we can determine the coefficients ai,a^,a^,.,, uniquely
in terms of a©, which shews that all expressions of the kind are multiples of
any one of them); and as the value found for ao is equal to the coefficient of
the initial term in the expansion of Qn (z\ we have
y^Qnizy
Thus we have the result
Qn W = [ {^ + (^* - 1)* cosh ^}-*-i d0,
which may be regarded as the analogue of the Laplace's integral already
found (§119) for Pn(^).
The theorem is valid only when the real part of (n + 1) is positive ; and
the proof has assumed that | ar | > 1 ; but the equivalence of Q„ (z) and the
integral, having been proved to subsist for this range of values of z, must
continue to subsist for all values of z, continuous with this range, for which
the integral continues to represent an analytic function of z ; and hence the
theorem holds for all values of z except those which are real and less than
— 1, which are singularities of the integral
126. Relation between Pn(z) and Qn(z), when n is integral.
When n is a positive integer, and z is not a real number between 1 and
— 1, the functions Qn{z) and Pn(z) are connected by the relation
which we shall now establish.
126] LEGENDRE FUNCTIONS. 227
When 1 ^ 1 > 1, we have
l/>«.4^-U>w^('-!-^^••■)•
Now if (n + A) is an odd integer, we have
r -P" (y) ^^y = f '-P- (y) y^ - ?p- <y) ^^y = «.
J -1 Jo Jo
If n is less than ifc, and (n 4- A;) is an even integer, we have
\ \\Pn (y) y'dy = l^n (y) y'dy
1 fi d^
(by Rodrigues' theorem) = g^j J^ y* ^^n (y' - 1)" dy
1 f^
(integrating by parts) = ^^ A; (A; - 1) . . . (Ar - w +1) y*"* (1 - ff dy
JL Til J
=2^!^'^*""^^^*""^)-(*-^-^^^^(^r^' ^ + i)
- A?(A;-1)(Aj-2)... (Jk-n + 1)
'"(fc + n+ l)(A; + ?i-l)...(A?-n+l)'
If on the other hand h is less than n, and (n H- &) is an even integer, the
same process shews that the integral vanishes.
Therefore
A;(A;-l)...(i-n4-l) 1
2J.i '^^^^i^-y"" (A: + n4-
1) (i 4. n - 1)... (A;-n+l) -e*+i '
where the summation is taken for the values A; = n, n 4- 2, 71 + 4, n + 6, ... 00 .
But this expansion, by § 123, represents Q,n{s^- The theorem is thus
established for the case in which |£:| > 1. Since each side of the equation
Qn(.) = |/;P„(y);^
represents an analytic function even when \z\ is not greater than unity,
provided z is not a real number between — 1 and 4- 1, it follows that, with
this exception, the result is true universally.
Example, Shew that Q« {z\ where n is a positive integer, is the coefficient of A* in the
expansion of(l-2A«+A*)-ico8h-i j- ^ " \ .
For
n=o 11=0 * 7 -I *— y
_1 n (l-2Ay+A«)~*d y
=(1 -Steft+A«)-*co8h-> {-f J J J •
16—2
228 TRANSCENDENTAL FUNCTIONS. [CHAP. X.
127. Development of the function (t^x)"^ as a series of Legendre
polynomials in x.
We shall now obtain an expansion which will serve as the basis of
a general class of expansions involving Legendre functions.
We have, by the recurrence-formulae,
(2n + l)a:Pn(a:)-(n+l)Pn+i(a?)-7iP,^i(a:) = 0,
(2n + 1) zPn {z) - (n 4- 1) Pn+i {z) - nP^i {z) - 0.
Multiply the firat of these equations by Pn{^\ the second by Pn(^)>
and subtract: we thus obtain
(2n + l){z^x)Pn{z)Pn{x)
= (n + 1) {Pn^, (Z) Pn {X) - Pn (z) Pn^i (x)}
- n {Pn (Z) Pr^, (X) « Pn (x) P^^ (z)].
Write n = 0, 1, 2, 3, ... n successively, and add the resulting equations.
This gives
{P,(x)P,(z)-^SP^(x)P,(z)+...-\-(2n-\^l)Pn{x)Pn(z)}(z-x)
= (n + 1) {Pn+i (Z) Pn (^) - Pn+1 (^) Pn (z)].
Divide throughout by (z — x){z — 1\ and integrate from -? = — 1 to
^ = + 1.
Thus
x\'^\2r^\)Pr{x)^^dz
^r., {.z-x)iz- i) ^^»^- <^> ^» ^"'^ ■ ^•^' ^''^ ^" ^^>' "^
(by partial fractions) = — — I -— — {Pn+i (-8^) Pn (^) - Pn+i (^) Pn (^)} d-^
"/I't^ {^»+i (^) ^» (^) - -Pn+i («') i^n (^)l d^ .
Now by the result of the last article, the left-hand side of this equation
can be written
-2l(2r-hl)P,(a;)e,(0.
In the first integral on the right-hand side, replace the integrand by its
n
value 2(2r H- l)Pr(a?) Pr{z), and integrate : only the first term survives, since
I Pr{z)dz^O,
when r is an integer greater than zero ; so the integral has the value 2.
127]
LEGENDRE FUNCTIONS.
229
We thus have
2 (2r + 1) P, (w) Qr (t) = -L. + }±l {P„ (x) Qn+, (t) -P„+. (x) Q„ (0).
This equation is valid for all values of n. Let us now see if x and t can
be so chosen as to make the last part of the right-hand side tend to zero as
n tends to infinity. We have, from Laplace's formulae for the functions
Pn and Qn,
p. w «.„ «) - p... (.) «. «) -If J] {i^i^r ^ #"*.
where A denotes a quantity which is finite and independent of n.
It is clear that this double-integral tends to zero only when, for all values
of (f> between zero and tt, and all values of -^ between zero and infinity,
the inequality
a;-|-(a^«--l)*cos<^
t-h(e*-l)*coshi|r
<1
^ = K^^i
t
'Ih^i)'
is satisfied.
Writing
the inequality becomes
uA viu — I cos<f>
u \ uj ^
The left-hand side of this relation has its maximum value when cos^s 1,
the value being 2 1 u |.
The right-hand side similarly has a minimum value equal to 2 { t; {.
The condition thus becomes
v^ h ft;r--jcosh'^
u\ < \v
or
|a:H-(^-l)*|<|t-h(<«-l)*|.
This inequality shews that the point x must be in the interior of an ellipse,
which passes through the point t, and which has the points H-l, — 1 for its foci :
for if a be the major axis of this ellipse, then
t = a cos + i (a* — 1)* sin 0,
where is the eccentric angle of t in the ellipse ; and thus
(t^ -. l)i = (a»-- 1)* cos ^-huisin 0,
and e + (^» - 1 )* = {a + (a« - 1)*} e^,
so that |^ + (^2-i)*| = a + (a»-l)*,
and hence the above inequality shews that the semi-axis of the ellipse which
passes through x is less than the semi-axis of the ellipse which passes through
t, i.e. that x is within the ellipse which passes through t.
230 TRANSCENDENTAL FUNCTIONS. [CHAP. X.
Hence if the point x is in the interior of the ellipse which passes through the
point t amd has the points + 1,-1, for its foci, then the expansion
-i- = I (2« + l)P„(a:)<2„(0
t — X n=0
is valid,
128. Neumann's theorem on the expansion of an arbitrary function in
a series of Legendre polynomials.
We proceed now to discuss the expansion of any arbitrarily given function
in terms of the polynomials of Legendre. The expansion is of special interest,
inasmuch as it represents the case which stands next in simplicity to Taylor's
series, among expansions in series of polynomiala
Let f{z) be any function, which is regular at all points in the interior of
an ellipse C, whose foci are at the points -^ = — 1 and z^ + l. We shall
shew that it is possible to expand f{z) in a series of the form
aoPo(^) + a^Pi {z) 4- a^P^{z) + a^Pj,{z) + ...,
where Oq, a,, a,... are independent of z : and that this expansion is valid for
all points z in the interior of the ellipse G,
For let £r = ^ be any point on the circumference of the ellipse.
Then we have
or /(^)= S anPn{z\
where an= gTri j /(OQn(0^-
This is the required expansion.
Another form for a^ can be obtained in the following way.
We have
_ 2n + 1
rV/<^>^^--2/>^<^>.-^
2n + 1 /•+!
2
2n + 1 r+i
r>'<^^^^\{-Bf
f_f{y)Pn{y)dy.
2
The latter is the more usual form for On-
128, 129] LBGENDRE FUNCTIONS. 231
Example 1. Shew that the semi-axes of the ellipse, within which the series
converges, are
K''+D*"'*K''-J)'
where p is the radius of convergence of the series
Example 2. If
Vy+l/ ' (^+l)(y-l)
prove that
129. The associated functions Pf!^{z) and Qn^iz)-
We shall now introduce a more extended class of Legendre functions.
If m be any positive integer, the quantities
will be called the associated Legendre functions X>f the nth degree amd mth
order, and will be denoted by Pn^(z) and Qn^(z) respectively.
We shall first shew that the associated Legendre functions satisfy a
differential equation analogotcs to the Legendre differential equation.
For let the Legendre differential equation
d^v
be differentiated m times, and let v be written for -r-^ .
We thus have for v the equation
(l--?»)^,-2ir(m + l)^ + (n-m)(nH-m + l)t; = 0.
m
Write w = v(l-2:*)2;
the equation becomes
.- -. dhv e, dw ( , , . mM ^
This is the differential equation satisfied by the functions
P„«»(^) and Q„'»(4
232 TRANSCENDENTAL FUNCTIONS. [CHAP. X.
Several expressions for the associated Legendre functions can be obtained
easily from the above definitions.
Thus from Schlafli's formula, we have
m
iiTr* Z J (J
where is a simple contour enclosing the points ^^1 and t^z, but not
enclosing the point ^ = — 1.
From this result, or from Rodrigues' formula, we have, when n is
a positive integer,
130. The definite integrals of the associated Legendre fimctions.
The theorems already given in § 115, relating to the definite integrals of
the Legendre functions, can be generalised so as to be stated in the following
form : When m and n are positive integers,
j Pn'^{z)P/^(z)dz^O, when r<n,
and r^P^''('^}'^'-A,?^r
J-i^ ^ 2n + l(n— m)I
To establish these results, we use the identity
which gives
L r (^ + >")' / nm
{^(^'-i)*H^<^- '>"}'''
(integrating by parts) = aS^^^J' jd^(^-l)f <i'
2 (n + m)!
2n+ 1 {n''m)V
i
130, 131] LEGENDRE FUNCTIONS. 233
We can prove in the same way the other result stated, namely that
I P„"* (z) Pr"" (js) dz = 0, when r + n.
For this integral in the same manner reduces to a multiple of
which is zero when n and r are diflFerent.
131. Expression of Pn^{z) as a definite integral of Laplace* s type.
The associated Legendre functions can be expressed by means of definite
integrals of the same type as those found in § 119 and § 125, as will appear
irom the following investigation.
We have
/ {z + cos (^ - !)*}«-*" sin^ (f>d<l>
Jo
= - {2r4-cos0(^>-l)*}**-^sin*^-^<^cos<^
+ 1 COS (f>-j-r [sin««-i (f>{z + cos <^ {z^ - 1)*}**-*^] d<f>
Jo a<f>
= (2m - 1) f 'cos« <f> sin«^-^ <l>{z+ cos (z^ - l)*}**"^ d(f>
Jo
- (n - m) ('{z + cos (z^ - l)*}*-«-i cos <^ (z^ - 1)* sin«~ <l>d<f>
Jo
= (2m - 1) f 'sin**^ (f>{z-\- cos ^ (z^ - l)l}»-«» d<^
- (2771 -1)1' sin»^ { J + cos <^ (^ - 1 )*}«-^ d^
- (71 - m) / (^ H- cos <^ (z^ - 1)*}*-^ sin«'~ <^d0
+ (n-m)z I f^r + cos {z^ - l)*}»-«-i 8in»» ^d^.
Jo
We thus have
(n + m) j {z + cos <^ (^ - l)i}»-»» sin*» <^d<^
= (2m - 1) Tsin^^-^ <f>{z + cos <^ (-g« - 1)*}*»-^ d<^
Jo
- ^^ ^ I {^ + cos (f> {f - l)*}~-« 8in»^^ <^1
+ ^^ I {-g: + cos <^ (-?» - l)*}»-*»(2m -1) 8in*^»<^ cos ^d<^,
234 TRANSCENDENTAL FUNCTIONS. [CHAP. X*
"^ 2m
?:±^ ['[z + cos A (^ - 1)*}'-^ sm«~ <^d<^
Zm — 1 J
= rsin««-« <^ {ir + cos (z* - !)*}*-« {1 + r (-r« - 1)-* cos <^} d<f>
Jo
= — — - -J- \{z + COS 6 (z^ - l)i}«-"»+i 8in«*-» <bd6.
n — m + 1 d^^Jo T-v /) Y' r
Thus if we write
Tm-l (2: + cos (^ - !)*}'*-« sin** <^c^,
we have /„=_-(2^1) %> ,
(n + m)(n— OT+1) dz
and therefore I - (2m- l)(2m-3) ... 1 ^
** (n + m)(n + m-l)...(n-m + l) cJ^**
But /o = f '{^ + cos <^ (^» - 1 )*}*» d<^ = irPn {z\
Jo
when the real part of z is positive.
Therefore /. = , (2m -1) (2m -3) 1.^ ,, ;^Pn(.)
= (2m-l)(2m-3)...1.7r ^ -^ ^
(n + m)(nH-m-l)... (n-m + l)^ '^ "^ ^ ^'
"""^ ^~ ^^^^ (2m-l)(2m-3)...1.7r ^^"^^
X I {-^ + COS <^ (^« - 1)*}»-^ sin*** ^ d^.
Jo
This result expresses Pn^(z) as a definite integral of Laplace's type, valid
for all values of n when the real part of ir is positive.
132. Alternative expression of PfJ^ (z) as a definite integral of Laplace* 8
type.
The formula last found can be replaced by another result, found in the
following way.
If in Jacobi's well-known theorem*
Jy (co8^)co8m^d^ = 1.3.5 ..\2m-l) /o'-^'"'^^^"*>°^°"'^'^-
we take /(cos ^) = {^ + (z^ — 1)* cos <f>}^,
* CreUe's Journal^ xv.
H
132, 133] LEGENDRE FUNCTIONS. 235
80 that
/w (cos <^) = n (n - 1) ... (n -m + 1) (z^ - 1)^ {2:+ (^ - 1)» cos <^}"^,
we obtain
I {^ + (^— l)*cos^}**cosm^d^
Jo
^ njn-l) ... (n-m-hl) , _ , .f
1.3.5...(2m-l) ^ ^
X I {z+ (z^- 1)* cos 0}»*-^ sin^ <f)d<f>
Jo
(n + m)(n +m— 1) ... (/i + 1)
Therefore
p^m /^\ ^ (^ + m)(n4-m-l) ... (n + l) , ^vf
TT
X 1 {2^ + (^ — 1)* cos (^J** cos m<f>d(f>.
Jo
This formula is valid for all values of w, and for all values of z whose real
part is positive ; m being a positive integer.
133. The function Cn*" («).
A function connected with the associated Legendre functions F^^ {z) is the function
On*" (^)) which for integral values of p is defined to be the coefficient of A** in the expansion,
in ascending powers of A, of the quantity
(l-2A^+A«)-»'.
It is easily seen that C^*" (z) satisfies the differential equation
<Py (2v + l)zdy n(n + Qv)
d^^ z^-\ dz z^-l ^""*
For all values of n and y, it may be shewn that C^ {z) can be defined by a contour-
integral of the form
Constantx (1 -««)*- ^ f ^lzf)-l^dt.
When n is integral, we have
Cy(z^= (-2)*v(i^4-l)...(v4-n-l)
* ^ ^ n! (27i+2v-l)(2»+2v-2)...(n+2i^)
which corresponds to Rodrigues' formula for P^ (z) ; in fact, since
F^(z) = Cj{z\
Bodrigues' formula is a particular case of this formula.
236 TRANSCENDENTAL FUNCTIONS. [CHAP. X.
When r is an integer, we have
«-'*^^^"(2r-l)(2r-3)...3.1 (£2^ *^^^'
whence we have
The last equation gives the connexion between the functions C^^ (z) and P/ (2).
This function C/ (z) has the following further properties, analogous to the recurrence-
formulae,
<t;(.)-c.(^)-£c;w=o.
Ow-<l«="-:^c'/(.).
r-i
Miscellaneous Examples.
1« Shew that when n is a positive integer,
p^{z)J-z:^-p(u^+z^r^,
where t^* is to be replaced by (1 -^) after the differentiation has been performed.
2. Prove that when w is a positive integer,
(Cambridge Mathematical Tripos, Part I, 1898.)
3. Shew that
• -1 w « fl.3.6...(2w-l)l«,„ .. „ ...
(Catalan.)
4. Prove that
is zero unless m- n«i ± 1, and determine its value in these cases.
(Cambridge Mathematical Tripos, Part 1, 1896.)
6. Shew (by induction or otherwise) that when n is a positive integer,
(2»+l)rP,*(«)(fc-l-*P,«-2i(P,«+i','+"+^»-t)+2(A'P»+^»^»+-+^»-i^«)
(Cambridge Matbematdcal Tripos, Part 1, 1899.)
/-
6. Shew that, if i{r is an odd number,
1
— jfe-2a»P»(4
(1-22A+A«)«
MISC. Exa]
LKOENDRE FUNCTIONa
237
where
8 a \i(t-»)
_A« 2^*-»(2n+l) / 8 3\
" (l-A«)*-«1.3.6...(it-2)V 8a? 8j^;
^ ^-i(2i»-t+4)yi (2n+Jfc-2)^
where a: and y are to be replaced by unity after the differentiations' have been performed.
(Routh.)
7. If
n-O
shew that
and
and
where
2 (n+1) 72»+i -3 (2n+l) /?^+(2n- 1) i2»_2=0
4(4a3-l)/J^'"+96««72»"-«(12n«+24w-91)i2^'-w(2n+3)(2n+9)i2»-0,
K-J^ , etc
(Pincherle.)
8. If m and n be positive integers, and m^n, shew that
P (z)P (z)== 2 A„,^rArA^.r ( 2n+2m-4r+l\
where
^«'
1.3.5... (2m- 1)
ml
(Adams.)
9. Shew that P^ (z) can be expressed as a determinant in which all elements parallel
to the auxiliary diagonal are equal (i.e. all elements are equal for which the sum of the
row-index and column-index is the same) ; the determinant containing (2n- 1) rows, and
its first row being
(Heun.)
_1 1 11
^ 3' 3''-"5' 5^'"*2w-l
t2.
10. Shew that
11. Shew that
2 r {z(l-i^-2t(l^m^ ^^^
12. Shew that, when n is a positive integer,
§^(cos^) (-1)** 8» fl
f^tn-*-!
n! 8«*
(2V*«(r-+i)}'
where ««=r cos ^.
(SUva.)
(Catalan.)
238 TRANSCENDENTAL FUNCTIONS. [CHAP. X
13. Shew that the complete solution of the Legendre differential equation is
y=^p.(.)-H5P.w/;^3-^^^,.
14. Shew that
{2 + (2i^l)i]a^ 2 5«§2m.«.lW,
where
„ o(a+2 w+i) r(m~^)r(m-a-^)
"*" 2tr ml r(m-a+l) *
•'x+(««-l]
dh.
15. Shew that, when the real part of (w+ 1) is positive,
and
/-^-(^t-i)* A**
16. Prove that
(Cambridge Mathematical Tripos, Part II, 1894.)
17. Shew that, if n be a positive integer,
18. Shew that
and
where n is a positive integer, and z > 1, and where log ^ is to be changed into log
z— 1 \ —z
if « is numerically less than unity.
Prove also that
V 2 3/ P223« V 2 / ■*"••*
whereit=l + - + 5+...+-.
2 3 n
(Cambridge Mathematical Tripos, Part II, 1898.)
MISC. EXa] LEGENDRE FUNCTIONS. 239
19. Shew that
-P«*»(«)=-^ ^-^, ^f^F{n, n+m, m+1, s^).
20. Prove that, if
^•"n (n«-l)(n2- 4) ...{»«-(«- 1)2} (w+«)^^ ^^ ^»
then
,. -P 2^2n + l) 2n+3
3 (271+3) p , 3(2n+5) ( 2n+3)(2n+ 5)
y3"^-+8 2«-l ^»+i"^ 271-3 *-i (2w-l)(2w-3)^'-5'
and find the general formula.
(Cambridge Mathematical Tripos, Part II, 1896.)
21. If
shew that
22. If
On"" {^^1 - (^ - 1)* (^1^ - 1)* cos <^}
_ n(2y-2) J^-* 4An(n-X){n(v+X-l)}g
{n(y-l)}2xfo^ ^^ n(n+2i.+X-l)
(Qegenbauer.)
<^» («) = I **(^- 3^2 + l)-i i»c?<,
where ^i is the least root of ^— 3te+ 1 =0, shew that
(27i+l)crn+i-3(2w-l)z(r„.,+2(n-l)(rn-2=0,
and
4(4«5-l)(r,,'"+1442j<r,»"-«(12w2-24n-291) <r,»'- (n-3) (2n-7) (2n+5) cr,»=0,
where
o-n' = - ^} , etc. (Pincherle.)
/'/
23. Shew that
(Hobson.)
g^n>(,)^^>i r(.^+l) r COShj^t.
r(n-w + l);o {r+(22-i)icoshw}-+i
where the real part of (n+l) ia greater than m.
24. The equation of a nearly spherical surface of revolution is
r=l+a{P,(cos^)+P3(co8^ + ... + P2n-i(cos^)},
where a is small ; shew that to the first order of a the radius of curvature of the
meridian is
l+a 2 {n(4wi+3)-(r?i+l)(8m + 3)}P8,n+i(cos^).
(Cambridge Mathematical Tripos, Part 1, 1894)
CHAPTER XL
Hypbrgeometric Functions.
134. The Hypergeometric Series.
We have already in § 14 cjonsidered the hypergeometric series*
a.b a(a + l)b{b + l) a(a + l)(a + 2)b(b + l)(b + 2)
■^l.c "^ 1.2.c(c+l) "^ 1.2.3.c(c-rl)(c + 2) "^ "'
from the point of view of its convergence. It was there shewn that the
series is absolutely convergent for all values of z represented by points in
the interior of the circle whose centre is at the origin and whose radius is
unity. It follows from §22 that all the series which can be derived from
the hypergeometric series by diflferentiation and integration are likewise
absolutely convergent within the same region : and by § 55, the convergence
is not only absolute but uniform over the interior of the circle, and the
sums of the series obtained by differentiation and integration of the series
term by term are the derivates and integrals respectively of the sum of the
series. The hypergeometric series, together with the series which can be
derived from it by the process of continuation (§ 41), will therefore represent
an analytic function of the variable z; this function will be denoted by
F(a,b,CtZ),
Many of the most important functions of Analysis can be expressed by
means of the hypergeometric series. Thus it is easily seen that
(l+zY^F(^n,/3,/3,^zl
log(l+5) = zi^(l,l,2,-5),
6^ = Limiti^U,/8,l,j),
and we have shewn in the preceding chapter that the Legendre functions
may be represented by the series
• The same was given by Wallis in 1655.
134, 135] HTPBRGEOMETRIC FUNCTIONS. 241
2-z-r(n-^i) (l^n n 1 1\
^«W- ^4r(n + l) ^\ 2 ' ""2' 2"^"' J«;'
O /.,x_ 7r>r(n + l) 1 y/n+1 7i + 2 3 1\
These examples are sufficient to shew that the functions represented by
the hypergeometric series are in some cases one-valued and in other cases
many-valued.
Example, Shew that
^^F(a, 6, c, «)=^F(a + l, 6+1, c+1, z).
136. FaZt^ o/* ^A« series F(a, 6, c, 1).
We have shewn in § 14 that the series F(a, b,c,l) converges absolutely
so long as the real part of c — a — 6 is positive. Suppose this condition to
be satisfied. Then we have
F(nhr^\-l. Tia + n)T(b + n)r(c)
F (a, 6. c, 1) ^ Z^ ^uT(c + n)r(a)r(6)
Tic) I 1 T{n + a)T{n-\-h)Vic-h)
r(a)r(6)r(c-6),tonI r(n + c)
r(c) I ir(» + o)5(n + 6,c-6)L
r(o)r(6)r(c-6),.on!
r(c)
= r(a)Tl)l(c-b) iy'^''^IS^' - ^y-'-^^'dz.
Writing z=^l — t, this becomes
(writing xt^s)
r(c)
r(a)r(6)r(c-6);o Jo
r(c)
zrr\j '^1 e-'8^^if>-<'-<>-'(l-tf-\dt
r(a)r(6)r(c-6)
r(o)5(c-a-6,6)
r(c)r(c-o-6)
~r(c-6)r(c-o)*
W. A. 16
242 TRANSCENDENTAL FUNCTIONS. [CHAP. XL
The hjrpergeometric series with argument unity can thus be expressed in
terms of Oamma-functions.
136. The differential equation satisfied by the hypergeametric series.
The function represented by the hypergeometric series y='F(a,b,c,z)
satisfies the differential equation
'i^-l)2^ + l-o + ia + b + l)z]^ + ab!, = 0;
for if the series be substituted for y in the left-hand side of the equation,
the coefficient of 2^ is
a(a + l)...{a + r-l)b(b + l)...(b + r-l)
1 .2 ...r.c(c + l)...(c + r)
{r(r'-l)(c-{-r)'-r(a-\-r)(b+r) -c{a+r)(b + r) + r(c + r)(a + 6 + 1) + oft (c + r)}
or zero ; which establishes the result.
Example, Shew that one integral of the equation
IS
where
^F{m-/i, m-v, m-n+1, «),
a-l--Oi+ir),
6+l=m+n,
137. The differential equation of the general hypergeometric Jimction.
The differential equation found in the preceding article is a case of a
more general differential equation, which may be written
d2l'
\ z — a z — b z — c ] dz
{ aa'{a^b){a-c) ^ /3^(b^a)(b^c) ^ yy-(c-a)(c-6) j y ^^
( 2r — a z — b z — c J (^— a)(2r--6)(2r— c)
* ...(A),
in which a, 6, c, a, ^, 7, a , ff, y are any constants such that the equation
a + /3 + 7 + a' + )S' + 7' = l
is satisfied. This will be called the differential equation of the general hyper-
geometric function. The form here given is due to Papperitz*.
* Math, Annaleuj zxv.
136, 137] HYPEEGEOMETRIC FUNCTIONS. 248
We shall now shew that the differential equation satisfied by the hyper-
geometric series is a particular case of this equation.
For in the equation (A), write
= 0, 6=00, c = l.
The equation becomes
y . f l-g-g' . I-7-7 ) dy ( ««' Tx' . oo/l _y_ n
In this equation, let a and 7 be replaced by zero. We thus have
d2^ \ z z—l J dz ziz—l)
and in this equation the constants ol, y, fi, ^, are to be such as to satisfy
the relation
/g + a' + ZS' + y^l.
This differential equation can be identified with the equation
z(z-l)^ + {-c + (a + b + l)z]^ + aby = 0,
which is the differential equation satisfied by the hypergeometric series, by
writing
y3 = a, )8'=:6, a' = l-c;
which in virtue of the above relation gives 7' = c — a — 6. The differential
equation of the hypergeometric series is therefore a special case of
equation (A).
We shall denote any solution of the general differential equation (A) by
the symbol
{a b c
a ^ y z}.
a! ^ 7'
This notation is due to Riemann* ; it enables us to express our result thus :
The hypergeometric aeries
F(a, 6, c, z)
is a solution of the differential equation of the class offu/nctions
t
00 1
P a z
[ 1— c b c—a—b
* Abhandlungen d, K, Getell, d, WUtemchaften zu OdtHngen^ Tn. (1857).
16—2
244 TRANSCENDENTAL FUNCTIONS. [CHAP. XI.
Although the hypergeometric series itself satisfies only a particular form
of the differential equation (A), it is nevertheless possible to satisfy the
general equation (A) by means of a function derived from the hjrper-
geometric function. For by the transformation
a? (5 — 6) (c — a) = (^r — a) (c — b),
the differential equation (A) is reduced to the form
cte* I X . a?— IJow? ( X a? — 1 } x{x — l)
In this we take a new dependent variable, defined by the equation
y = x*{l'-x)y u.
The equation becomes
, /I -a' + a , 1 -7' + 7\ du . ,^^ . x/i ^ /\> ^ /v
Now the equation
a + /S + 7 + a' + ^' + 7'sxl
will be satisfied if j3, a\ ffy 7', are expressible in terms of three new constants^
a, 6, c, defined by the formulae
/ /8 = a — a — 7,
a' = 1 — c + a,
^ = 6-a-7,
7'=sc — a — 6 + 7.
The differential equation for u can now be written
d^u du
x(x- l);i-i + {(1 +a + 6)a?-c}-Y- + a6M=0.
But this is the differential equation satisfied by the hypergeometric
series, a solution of it being
F (a, 6, c, x).
Hence we have, as one solution of the equation,
u^F(a + /3 + y, a + /9' + 7, 1 + a - < a:),
or y^x*(l- x)yF(a + ;8 + 7, a + /S' + 7, 1 + a - a', x),
or, disregarding a constant factor,
1
• 138] HTPERGEOMETRIC FUNCTIONS. 245
This is therefore a solution, expressed by a hypergeometric series, of the
differential equation which defines the class of functions
Ia b c
a ^ y z
The advantage of the differential equation (A) over the equation found
in § 1 36, which is satisfied by the hypergeometric series, lies in its greater
symmetry and generality. The points 2r = a, -e = 6, and z=^c, are called the
singularities of the differential equation (A); the quantities a and a' are
called the exponents at the singularity a; and similarly /8 and ^ are the
exponents at h, and 7 and y' are the exponents at c.
Example, Shew that
00 1 ^ / -1 00 1
fi y zn^p\ y 2/3 y
k »' y' ] I y s/y y
(Riemann.)
This relation follows from the fact that the differential equation corresponding to either
of the P-fiinctions is
138. The Legendre functions as a particular case of the hypergeometric
function.
The expressions which have been found for Pn(^) and Qn(^) as hjrper-
geometric series naturally lead us to suppose that Legendre's differential
equation is a special case of the differential equation which defines the
general hypergeometric function. That this is the case appears from the
following investigation.
If in equation (A) of the last article we take
a = — 1, 6 = 00, c = l,
we obtain the differential equation
d»y ( l.,a^g^ 1-ry^ryO dy r 2ga^ 277M y
d?^[ 5 + 1 ^ 5-1 ]dz^\ 5+l'^^^"^5-l](5-l)(5 + l) ''•
If now in this equation we take a = 0, a' = 0, 7 = 0, ♦/ = 0, yS = n + 1,
fi'ss^n, we obtain
or (l-^')§-25^ + n(n + l)y = 0,
which is the Legendre differential equation.
246
TRANSCENDENTAL FUNCTIONS.
[chap. XI.
It follows from this that any solution of Legendre's equation is a hyper-
geometric function of the type
-1 00 1
p\ n + 1 z
-n
In the same way it can be shewn that the associated Legendre
functions Pn^{z) and Qn^{z) are hypergeometric functions of the type
P\
-1
00
2 " + 1
1
m
\
I
m m
"2 -" "2
Example 1. Shew that
S^-«=^
-1 00
— r w+r+1
-n^r
1 \
-r z
• •
Example 2. If «^si7, shew that the Legendre difierential equation takes the form
c&,«"^\2,,"l-i7Jrfi7"^ 47(1-1;)
Shew that this is a hypergeometric differential equation.
139. TransformcUioris of the general hypergeometric function.
We shall next consider the effect of performing certain transformations
in connexion with the general hypergeometric function
^a b c
The differential equation satisfied by this function is
d^ \ z^a z — b z — c)dz\ ~ '
z — a
P^{b^a){b^c) . 77'(c-a)(c-6)
z^b
z — c
\ y_
){z- a)(z -
b)(z - c)
= 0.
In this equation, let the dependent variable be changed by the trans-
formation
139]
HTPERQSOMETRIC FUNOTIONS.
247
\
The differential equation for y' is found after a slight reduction to be
<fy ^ f l-«-«'-28 ^ l-/3-/y + 28 1-y-y' ) dy"
dz* \ z — a z — b z — c)dz
^ f (a + 8)(a' + 8)(a
1 z ^a
- 6)(a - c) ^ (/3 - 8)(/y - 3)(6 - c)(6 - a)
+ 77
,(c~a)(c-6)
z — c I (-P — a) (^ — b)(z — c)
^-6
= 0.
This is the differential equation of a hypergeometric function which has
exponents a + S, a' + S, at the singularity a, and exponents )8 — S, )8' — S', at
the singularity 6 ; and so we have
C-5
-./T.\«
a"
= P-
a 6 c
a + S )8— S 7 z
a' + S /y-s y
and hence in general we shall have
(-;)'&:)■' I
a b c .
a ff 7
= P
a
a + S id-
a' + S /3'
b c
-S— € 7+6
— S — € 7' + €
It will be observed that by this transformation the exponent-differences
a — a', )8 — )8', 7 — 7' are unaltered.
Consider now the effect of transformations of the indepeAdent variable s.
If we introduce in place of jer a new variable z\ defined by the equation
5=
Oi^ + ^i
C^ + di'
where Oifb^ Ci, di are constants, so that
we have
, - di^r + 6,
z =
and
dy _ Oidi - 61C1 dy ^ (c^z + d^ydy
dz (CiZ — Oi)* dz' Oidi — 6iCi de'
rf^*^ ((hz-ihT dz {CiZ-OiY d/*
^ 2c(c,z' + d,y dy {ci^ + d^Y d^
248
TRANSCENDENTAL FUNCTIONS.
[chap. XL
Hence if we define quantities a', b\ c by the relations
__ _ _ OiC + 61
Cia' + di * Cjjb' + di *
so that
the general hypergeometric differential equation becomes
[9. . (l-a-aO(Cxa^ + d,) , (1 -/3 - ^yXc^ + d,)
z — a
+
a-7"-70(
77'(o'"-a')(c'-6'))
y
7T7T7-:7x=0.
The coeflBcient of -7^ in this equation can be written in the form
l-.a-a^ l-/8~)y 1-7- 7 , 1
which, in virtue of the relation
2ci-(l-a-a')Ci \
reduces to
a + a' + /3 + )8' + 7 + y = l,
i^g^g- i^ff^/y i-7^y
Hence the differential equation reduces to the differential equation of
the function
a c
p\a fi y z
a' /3' 7'
and thus we have the relation
a 6 c
P-la fi y z
a' ^ 7'
o
=P-
6' c'
^ 7 ^'
^ 1'
This shews that the general hypergeometric function is unaltered if the
quantities a, b, c, z are replaced by qualities a', b\ c\ z^, which are derived
from them by the sam^e homographic transformation.
140] HYPERGEOMETRIC FUNCTIONS. 249
140. Ths twenty-four particular solutions of the hypergeometric differential
equation.
We have seen in § 137 that a particular solution of the general hyper-
geometric differential equation is
<z
^r C-^)V f ./..,... /r ^ ,-, 1 .. - .'. I^^fe^j)} .
We shall suppose that no one of the exponent-differences a — afy — ^,
7 — 7' is zero : it is shewn in treatises on Linear Differential Equations that
when this exceptional case occurs, the general solution of the differential
equation involves logarithmic terms; the formulae will be found in a
memoir* by Lindelof, to which the reader is referred.
Now if a be interchanged with a', or 7 with 7', in this expression, it must
still satisfy the differential equation, since the latter would be unaffected by
this change. We thus obtain altogether four expressions for which
(c — 6) (^ — a)
(c — a) (-8^ — b)
is the argument of the hjrpergeometric series, namely
.fz-ay' /z-c\y „{ , r, , OP / -, . / (c - 6) (-gr - a)'
these are all solutions of the differential equation.
Moreover, the differential equation is unaltered if the quantities a, a', a
are interchanged respectively with ^, ff, 6, or with 7, 7', c. If therefore we
make such changes in the above solutions, they will still be solutions of the
differential equation.
Let a change in which (a, a', a) are interchanged with ()8, /S', 6) be
denoted for example by
/a, 6, c\
U, a, 0) '
each singularity in the bracket being interchanged with the singularity
above or below it. Then there are five such changes possible, namely,
(a 6 c\ fa h c\ (a h c\ (ah o\ (a b c\
\b c a)* \c a bj* [a c b) ' [c b a)' \b a cj'
* Acta Soe, Scient. Fennieas, xix. (1898).
250
TRANSCENDENTAL FUNCTIONS.
[chap. XL
To each such change correspond (by interchanging a with a, etc. as already
explained) four new solutions of the differential equation. We thus obtain
twenty new solutions, which with the original four make altogether twenty-
four particular solutions of the h3rpergeometric differential equation, in the
fqrm of hypergeometric series.
The twenty new solutions may be written down as follows :
-c)j
+y8-)8',
+ /3'-/9,
+ /3-/9',
+ /8'-/8.
+ 7-7',
(a - h) (z
{a-c){z-h)\
(a-b)(z-oX
(a -c)(z — b)
-6)1
-c)\
(a - b) (z
(O — C) (^r — 6)
(6-
(6
- o) (g - c) |
- c) (^ - a)j
+ V-~ (^-»)(^-g) t
+ 7 7. (6_c)(^_a)[
(6 — o) (« — c)
+ 7-7'.
+ 7'-7»
(6 - c) (^ - o),
(6 — a)(^ — c)
(fc_c)(^-aX
„_„' (fe-c)(^-a ))
"' (6-a)(^-c)J
(6-c)(ir-a) ]
' (6-a)(^-c)J
Q>-c){z-a) \
(6-a)(^-c)[
(6-c)(^-o) ]
(6 — a) (5 — c)
(a-c)(-?
(a-6) (£-
Xa — c^iz
+ 'y "y' {fl-c){z-b)\
(a-6)(g-c))
-6)f
+ a-a',
+ a'-a,
+ 7-7'.
+ 7' -7.
-6)1
+ y'-%
(a — c)(z
141] HYPERGEOMETRIC FUNCTIONS. 251
The existence of these twenty-four values was first shewn by Kummer*
Example. Find the twenty-four solutions of the L^endre differential equation,
corresponding to the above set of solutions of the hypergeometric differential equation ;
and express each of them in terms of the two independent solutions P^ {z) and Q^ (z),
141. Relations between the particular solutione of the hypergeometric
diferential equation.
Since the twenty-four expressions found in the last article are solutions
of the same linear differential equation of the second order, any three of them
must be connected by a linear relation with constant coefficients.
We proceed to find the relations which thus connect them.
First, consider the set of four solutions
Vi* Vii Viz* yi6>
it is clear that, in the neighbourhood of the point z=^a, each of them can be
expanded in a power-series of the form
4 (2r - a)* {1 -h £ (^r - a) + (7(z - a)» + ...}.
But there is only one series of the form
(z-a)* {1 -h£ (^r- a)-f C(2r- a)» + ...}
which satisfies the differential equation; for the coefficients 5, C, ... can be
uniquely determined by actual substitution in the differential equation. Let
this solution be denoted by P<*>.
Thus the solutions
t/u Vty yi8, yi5
must be mere multiples of P^*'. Moreover,
for y, the factor A \s (a - c)y {a - 5)-c+y> ;
for y, it is (a - c)y\a - 5)-<«+y'» ;
for y^ it is (a — 5)^ (a — c)~*""^ ;
and for y„ it is (a — bY{a — c)~*~^'.
* Crelle*s Journal^ xv.
252 TRANSCENDENTAL FUNCTIONS. [CHAP. XI.
Thus we have
^ ^ Ka — cl xa — hl
xi-ja + ZS-K,. « + ^' + y. l-,„-a'.J--|(i^)}
\a — c' \a —
- 6\-"-r'
6,
x^{«+^+y, „+^+,. !+«-«, ;::^;;;:g }
=(-«)• (src^D
xP|a + ^ + -y. a + /9+r. l + a-«. (^ _ a) (^ - c) ,
■?■ fz-h\l>'
=<->-c^r(^^)
x'h^*''. ■>+«+»', 1+.-.', fti:^';::} }-
Similarly solutions P<*\ P<^», P<^\ P^y\ P<y'> exist, each of which is equi-
valent to four of the above hypergeometric series.
Having thus classified the twenty-four solutions into six distinct solutions,
namely
p(a)^ p(e')^ pO)^ p(/r)^ p(y)^ p(y')^
we proceed to find the relations between these latter six solutions. We know
that P'*> must be expressible linearly in terms of P<y> and P^y'K Let the
relation between them be
P(«) = a^p(y> + Oy^pty).
We have then to find the coe6Bcients Oy and oy.
Now this equation can be written in the form
r{'*»^i. '*g*i: '+«-«'. I^^Jl
z - 5\-»-r
-('-)' (j^ro
f |.^^^„ ^,^^„ ,^,.,., («_^|g^i
5-6N— *'
142] HYPERGEOMETRIC FUNCTIONS. 253
Dividing throughout by the common factor (z — a)*, and writing ^ = a and
z=sc successively in the resulting equation, we obtain two equations, from
which 7y and oy can be found : the hypergeometric functions reduce to the
type
F(u, V, w, 1),
which in § 135 was shewn to be expressible in terms of Gamma-functions, and
the type F {Uy v, w, 0), which clearly has the value unity.
As already explained, in certain cases (e.g. when one of the exponent-differences is an
integer) the above theory of the solutions requires modification. For a discussion of these
cases the student is referred to Lindel6f s paper already mentioned, and Klein's Lectures
" Ueber die hypergeometrische Function."
142. Solution of the general hypergeometric differential equation by
a definite integral.
We next proceed to establish a result of great importance, relating to
the expression of the hypergeometric function by means of definite integrals.
Let the dependent variable y in the differential equation of the general
hypergeometric function ((A) of § 137) be replaced by a new dependent
variable /, defined by the relation
y^iz-aYiz-hf (z-c)y I.
The differential equation satisfied by / is easily found to be
1 4-/9-/8' . 1 + 7-7]
d*/ fl+a-a l4-i8-/8'
dz^ ( z^a z — o
z — o
dJ
dz
(a + /3 + 7){(tt4-/3 + 7 + l)^ + Sa(a + /y-fy-l)}
"^ {z-a){Z''b){Z'-c)
which can be written in the form
i
(LJMi_l) p. (^) + (f _ 1) ^> (^)
/.
where { f = 1 -a-/3-7 = o' + /3' + 7',
Q{z) = {z-a)iz-h){z-c),
R(z) = 2 (a + y3 + 7) (z -b)(z- c).
It must be observed that the function / is not regular at e.'oo , and consequent! j the
above differential equation in / is not a case of the generalised hypergeometric equation.
We shall now shew that this differential equation can be satisfied by an
integral of the form
254 TRANSCENDENTAL FUNCTIONS. [CHAP. XI.
provided the path C of integration is suitably chosen.
For on substituting this value of / in the differential equation, the
condition that the equation should be satisfied becomes
= f (^ - ay+P+y' (t - 6)*+^'+r-i (t - cy-^P+y'-i {z - ^)-«-^-y-2 xdt,
J c
where
+ {t-z)[R{z) + {t-z)R{z)]
= {i-^)[Q(t)-{t-zy\ + it-z){R(t)-{t-zr^(<^ + p + i)]
^{^-^)Q{t) + {t-z)R{t)
= -(l + a + /3 + 7)(<-a)(<-6)(<-c)
+ 2(o' + /3 + 7)(<-6)(«-c)(«-«),
or Z = (< - a)'-"'-*-y (« - hy-^-^-y (t - cf-^-f-y' (z - <)«-H>+y+«
^ {(« - aY'+^+y (t - by+^'+y (t - c)'+p*y' (t - ^)-o+«+p-h')).
It follows that the condition to be satisfied reduces to
dV
o-lis"-!/^-
where F= {t - a)*'+^+y {t - 6)«+^'+> (< - c)*+^+>' (^ - 5)-ti+«+^+y).
The integral / will therefore be a solution of the differential equation,
provided the path of integration C is such that the quantity V resumes its
initial value after describing the arc C.
Now F= (t - ay-^^-^r-^ (t - 5)*+^'+>-i {t - c)»+^+>'-^ (z - 0"^"^"^ ^,
where U^it-a) (t- b) (t - c) (^ - 1)'^ ;
and the quantity U resumes its original value after describing any contour :
hence if (7 be a closed contour, it must be such that the integrand in the
integral / resumes its original value after describing the contour.
Hence finally any integral of the type
{z^aY{z^hy{z''C)y f (t-ay-^-^^'-'(t-b)y-^'^P''Ht-cY-^-^'-'(z-t)'^-fi-ydt,
J c
142] HTPERGEOMETRIC FUNCTIONS. 255
where C is either a closed contour in the t-plane such that the integrand
resumes its initial value after describing it, or else is an arc such that the
quantity V has the same value at its termini, is a solution of the differential
equation of the general hypergeonietric function.
Example 1. As an example, we shall now deduce a real definite integral which (for a
certain range of values of the quantities involved) represents the hypergeometric series.
The hypergeometric series F(a, by c, z) is, as already shewn, a solution of the differential
equation of the function
' 00 1
P ■ a t
1-c b c^a-b
The int^ral
thus becomes in this case
I <«-«(<-l)«-*-i («-«)-« eft.
Now the quantity F is in this case
and this tends to zero at ^b 1 and «= oo , provided c> b>0.
Hence if these conditions are fulfilled, we can take as the contour C an arc in the
^-plane joining the points t^l and <nao ; so that a solution of the differential equation is
/
00
f^ if- l)«-*-i {t-z)-^dt.
1
In this integral, write t^-; the integral becomes
/
this integral is therefore a eolutum of the differential eqtiatumfor the hypergeometric series.
It is easily seen that this integral is in fact a mere multiple of the hypergeometric
series
jP(a, 6, c, z) ;
for supposing | « | < 1, and expanding the quantity (1 - uz)'^ in ascending powers of z by
the Binomial Theorem, the integral takes the form
fV»(l-«)'-^»du+ i «(«+l)-(«+*-l) ^[V-'*r(i_^)«-t-.rf,^
Jo r=l T' Jo
or
r«i ri
256
TRANSCENDENTAL FUNCTIONS.
[chap. XL
or
or
nth . m/io. ; a{a'\-l)''>{a+r-\)b{h+\).,.{h'\-T-\) \
a(a+l)...(a+r-l)6(6+l)...(fe-hr~l)
r!c(c+l)...(c+r-l)
5(6, c-6)jP(a,6, c,z),
which establishes the result stated.
Example 2. Deduce SchUiflUs integral for the Legendre functions^ as a case of the
general hypergeometric integral.
Since the L^;eQdre equation corresponds to the hypergeometric function
-1 00 1 ^
n+1 £ V,
. -n J
the corresponding integral is
or
[ (^-l)«(«-0)-'»-icfe,
taken round a contour C such that the integrand resumes its initial value after describing
it ; and this is Schlafli's int^ral.
Example 3. Deduce Laplace's integral for the Legendre functions, as a case of the
general hypergeometric integral.
If we write
«=i(«*+r*),
4
the Legendre differential equation becomes
^-i-P I ^ \^y n(ti+l) y
This corresponds to the hTpergeometric function
.0 00 1
n
"2
n + 1
2
f
- y
n+1
2
n
2
1
and so the hypergeometric integral becomes in this case
("i I tt» ( 1 - u) "* (i - w)"* du,
taken round a contour enclosing the points u=l and u^(.
Write
Then the integral becomes
taken round a contoiu* enclosing the points u^l and u^CK
143] HYPERGEOMETRIC FUNCTIONS. 257
Write u=sA( in this integral ; we thus obtain
(l-.2«A+A«)-U»cM,
/'
the integral being now taken round a contour in the A-plane enclosing the points k=( and
Suppose now that the real part of z is positive ; and let the contour become so attenuated
as to reduce to a small circle surrounding the point h=(^ another small circle surrounding
the point h=(~\ and the line joining the points ( and f "^ described twice. The small
circles contribute only infinitesimally to the integral, which thus becomes a multiple of
/
^^(1-2M+A«)-U«(i^
Writing A =«+(«•- 1)* cos
in this integral, we obtain
/;
{«+'(««-l)*C08<^}*rf<^,
which is one of Laplace's integrals (§ 119).
143. Determination of the integral which represents P<*^
We shall now shew how the integral which represents the particular
solution P<*^ (§ 141) of the hypergeometric diflferential equation can be
found.
We have seen (§ 142) that the integral
7=(^-a)«(^-5)^(^-c)vf (^-a)^+y+«'-H«-6)y+*+^'-H«-c)*'^^'-K'^-0"*"'^"^*
J c
satisfies the diflferential equation of the hypergeometric function, provided
(7 is a closed contour such that the integrand resumes its initial value after
describing C, Now the singularities of this integrand in the ^plane are the
points a, b, c, z; and on describing a simple closed contour enclosing the
singularity 6 alone, the integrand resumes its initial value multiplied by
I Jiniy+a+fi'-l)
as is seen by writing it in the form
(^+y+tt'-l)log«-a)+(y+a+/3'-l)log(f-6)+(«+/8+y-l)log(<-c)-(a4-/3+Y)log(«-0
e .
Take then a point in the ^-plane, and draw a loop in the ^-plane passing
through and encircling the point 5, but not encircling any of the points
a, c, z. Let an integral taken in the positive or counter-clockwise direction
of circulation round the perimeter of this loop be denoted by the sign
Jo •
W. A, 17
258 TRANSCENDENTAL FUNCTIONS. [CHAP. XI.
and let an integral taken in the negative direction of circulation round the
perimeter of the loop be denoted by
r(6-)
so that we have the equation
/•(&+) r(b-)
Jo Jo
where it is understood that the initial value of the integrand in the second
integral is taken equal to the final value of the integrand in the first
integral
Let now a contour C be drawn in the following way. Take first a loop
starting from 0, encircling the point 5 in the positive direction, and returning
to ; then a loop starting from 0, encircling the point c in the positive
direction, and returning to 0; then a loop encircling the point 5 in the
negative direction ; and lastly a loop encircling the point c in the negative
or clockwise direction.
Conformably to the notation already explained, an integral taken round
this contour will be denoted by
{b+, c+, 6-, C-)
Jq
Now after description of this contour, the integrand of the integral /
already considered resumes its initial value multiplied by
gaIrf(y+tt+/8'-l-hE+^47'-l-y-«-^'+l-«-^-y'+l)
or 1, i.e. the integrand resumes its initial value*.
Hence if C be taken as the contour, the integral / will satisfy the
differential equation.
Thus
/•(6+,c+.6-.c-)
J = (^ - aY (z -hfiz" c)y (t - ay+y +«'-' (t - 5)y+-+^'-^
Jo
satisfies the differential equation of the hypergeometric function.
Now suppose that the point z is taken near to the point a, so that |z — a|
is less than either |6 — a| or |c — a|. We can clearly draw the contour just
* These double-circuit inUgraU were introdaced by Jordan in 1887. Clearly any namber of
contours can be formed in this way, it being necessary only to ensare that each singular point is
encircled as often in the negative or clockwise direction of circulation as in the positive or counter-
clockwise direction.
144] HTPEBQEOMBTRIC FUNCTIONS. 259
described in such a way that, for all points ^ on it, I ^ — a | is greater than | ^ — a | .
Thus we can write
Jo
(. . ^-, (, - izf)
z - a\-*-^-y
dt
Under the conditions already stated, each of the expressions
can be expanded by the Binomial Theorem in ascending powers of (z — a).
We thus obtain for I an expansion of the form
/ = (2r - a)« {4 + £ (ar - a) + (7 (^ - a)« + . . . },
and as / satisfies the differential equation it must therefore be a multiple of
the particular solution P**^ of § 141.
Thus
pw = Constant x (^ - a)« (z -hfi^z- c)^ {t - af-^y^'"^
Jo
(t - 6)y+«+^ -> (t - c)-^+y'-i (z - 1)-^-^-^ dt
Similarly
r(6+.c+,6-,c-)
P^*') = Constant x (z - a)*' (z ^hf{z- c)y {t - ay+y+— ^
In the same way the particular solutions P^\ P<^'»,*Pw, P<y'>, can be
expressed as contour-integrals.
144. Evcduation of a dovhU-contour integral.
We may note that an integral
/;
can be expressed in terms of the integrals
in the following way.
Let the initial value of the integrand at the point be denoted by T. After describing
the loop roujid a, the integrand will have at the value g^'^^^+^+V"^) Ty and the part
17—2
■(o-»-,6+, o-,6-)
260 TRANSCENDENTAL FUNCTIONS. ' [CHAP. XI.
I of the integral I will have been obtained. Describing next the loop
/(o+, 6 + . 0-, 5-)
will therefore be
^»i(a+/r+y-
■'/r*''
and the integrand will return to with the value
^ (tt'+/8+y-l+e+^'+y-l) qt
Describing next the loop round a in the negative direction, we observe that the corre-
sponding part of the integral would have been
r:
if the integrand had had for initial value
which is its final value when the loop is described with the initial value ^ : it is therefore
actually
or — e
jo '
i. '
_ 21^i(«+^'+y-l)
and lastly, describing the loop round h in the negative direction, we obtain the part
of the integral.
Collecting these r^ults, we have
/■''•+'*+'»-'*-L(i-«««t«+c'+y-i>) /■'"*'_(! _««'rt(«+o'+y-i)) /■•***,
a formula which furnishes the value of the double-contour integral in terms of two simple-
contour integrals.
146. Relations between contiguous hypergeovietric functions.
Let P (z) be a hypergeometric function with the argument z, the singu-
larities a, 6, c, and the exponents a, a',/8, I3',y,y. Let Pi+i^rn-i(z) denote the
function which is obtained by replacing two of the exponents, I and m, in
P(z) hy I + 1 and m —1 respectively. Such functions Pi+i^fn^i{z) are said
to be contiguous to P (z). There are clearly 6 x 5 or 30 contiguous functions,
since I and m may be any two of the six exponents.
It was first shewn by Riemann* that the function P(z) and any two of
its contiguous functions are connected by a linear relation, the coefficients in
which are polynomials in z,
»
• Abhandlungen der Kdn, Qet. der Wiss. zu O&ttingen, 1867.
145] HTPERGEOMETRIC FUNCTIONS. 261
80 29
There will clearly be — ^ — or 435 of these i*elations. In order to obtain
them, we shall take P (z) in the form
P (z) = (^ - a)* (^ - by (z - c)y lit" df-^-^'-^ (t - 5)y+«+^'-i
Jc
(t - cy+^+y'-i (t - ^)— ^ dt,
where C may be any closed contour in the ^-plane such that the integrand
resumes its initial value after describing C,
First, since the integral round C of the differential of any function which
resumes its initial value after describing G is zero, we have
= J ^ {(^ - ay-^f"^ (t - 6)-+/»'+r-i (t - c)«+^+y-» (t - r)-H»-y} dt,
or
= (a + ^ + 7) f (« - ay-^+r-' (t - ftV+^'+y-i (^ - cy-^P+y'-i (t - -?)-- ^-r dt
Jo
+ (a + )8' + 7 - 1) f (« - a/'+^+T' (e - 6)*+/»'+y-« (« - c)*+^+r'-i (t - «)-«H»-y (ft
+ (a + )8 + 7' - 1) f (t - a)*'+^+y ft - 6)•+^'+r-l rt - cV+^+y-i (^ - ^)-Hi-y d^
- (a + )8 + 7) f (« - a)*'+^-h' (e - 6)*+^'+y-» (t - c)-^^+r -^ (« - ^)-Hi-y-i d«,
or
(a' + )8 + 7)P + (a + )8' + 7-l)P.'+i.r-., + (a + )8 + 7'-l)P.'+,.y-i
(a-h^ + 7) p
^-6
/l+l.Y'-l.
Considerations of symmetry shew that the right-hand side of this
equation can be replaced by
jer — c
These, together with the analogous formulae obtained by cyclical inter-
change of (a, a, a) with (b, /8, )8') and (c, 7, 7O, are six linear relations
connecting the hypergeometric function P with the twelve contiguous
functions
P«+l.A'-l» P/M-1,y'-1> Py+1,«'-1> Ptt+l,y-l> Pi8+1,«'-1, Py+l.A'-l*
Pa'+i.ir-i, P«'+i.y-i, P/i'+i^y-i, P/i'+i,.'-!, Py+i,.'-!, Py+i,/i'-i.
262 TRANSCENDENTAL FUNCTIONS. [CHAP. XI.
Next, writing < — a = (^ — 6) + (6 — a), and using Pa'-i to denote the result
of writing a' — 1 for a' in P, we have
Similarly P = Pa'_i,y+i + (c - a) P.-i.
Eliminating P.'-i from these equations, we have
(c - 6) P + (a - c) P«'-i. fi^+, + (6 - a) Pa'-i.y+i = 0.
This and the analogous formulae are three more linear relations con-
necting P with the last six of the twelve contiguous functions written above.
Next, writing (^ — jer) = (^ — a) — (jer — a) we readily find the relation
P = ^j P/1+i.y.i - (^ - aY^' (z - b)f^ (z - c)y
xf(t- ay-^-^'-^ (z - a)y+«+/»'-^ (z - 6)«+^+y-i (^ - ^)-«H»-y-i d^,
which gives the equations
{z - a)-» {P - (^ - 6)-^ J'/i-M.y-il = (^ - 6)-^ [P-i^' or' Pr+i..'-il
- (^ - c)-^ {P - (^ - a)-> P.-M.^'-i}.
These are two more linear equations between P and the above twelve
contiguous functions.
We have therefore now altogether found eleven linear relations between
P and these twelve functions, the coefficients in these relations being rational
functions of z. Hence each of these functions can be expressed linearly in
terms of P and some selected one of them ; that is, between P and any two of
the above fwnctions there exists a linear relation. The coefficients in this
relation will be rational functions of z, and therefore will become polynomials
in z when the relation is multiplied throughout by the least common multiple
of their denominators.
The theorem is therefore proved, so far as the above twelve contiguous
functions are concerned. It can in the same way be extended so as to be
established for the rest of the thirty contiguous functions.
Corollary, If functions be derived from P by replacing the exponents
a, a', )8, )8', 7, </, by a+p, «' + ?, )8 + r, /S^H-*, 7 + ^, 7 +ti,where p, 5, r, «, f, w,
are integers satisfying the relation
then between P and these functions there exists a linear relation, the co-
efficients in which are polynomials in z.
MISC. BXS.] HYPERGEOMETRIC FX7NCTI0NS. 263
This result can be obtcdned by connecting P with the two functions by
a chain of intermediate contiguoi^s functions, writing down the linear
relations which connect them with P and the two functions, and from these
relations eliminating the intermediate contiguous functions.
It will be noticed that many of the theorems found elsewhere in this
book, e.g. the recurrence-formulae for the Legendre functions (§. 117), are
really cases of the theorem of this article.
MlSOELLANEOUS EXAMPLES.
1. Shew that
c
2. Shew that
jP(a+l, 6+1, c, t)-F(a, 6, c, t)^^F{a+l, 6+1, c+l, a).
3. If P(«) be a hypergeometrio function, express its derivates -^ and -^ linearly in
terms of P and contiguous functions, and hence find the linear relation between P, -^ ,
cPP
and -rj , i.e. verify that P satisfies the hypergeometric dififerential equation.
4. If
W(a,b,x) denote ^(-^i h 2, -bx\
shew that the equation y= TF(a, 6, a?)
is equivalent to a?-« Tr(6, a, y).
5. Shew that a second solution of the differential equation for
F (a, 6, c, s)
is a;»-«F(a-<J+l, 6-c+l, 2-c, x).
6. Shew that the equation
(«2+V)^+(«i-»-V)^+K-»-Wy=o
can, by change of variables, be brought to the form
and that this latter equation can be derived from the hypergeometrio equation
by the substitution 6^^, x— , where m is infinitely large.
264
TRANSCENDENTAL FUNCTIONa.
[chap. XI.
7. Shew that
o:(z)^p
/ -1 00 1
I
— «
where C («) is the coefficient of A*» in the expansion of (1 - 2A«+A*)-«' in ascending powers
of A.
8. Shew that, for values of x between and 1, the solution of the equation
where il, ^, are arbitrary constants and F{af /3, y, x) represents the hypergeometric series.
(Cambridge Mathematical Tripos, Part I, 1896.)
9. Shew that the differential equation for the associated Legendre function Pi!!*^{z)
of order n and degree m is satisfied by the three functions
(
00
\
1
2
p \ « W* ""^ — w
1 1-2
2^ ^"
v-s"^ n+1 -^m
00
P\
n
~2
n-hl
V 2
2 «-(««-l)* V,
— m
n+1
/ 00 1
P\
n m - 1
2 2 " l3^
n+1 9n 1
V"T" "2 2
10. Shew that the hypergeometric equation
(Olbricht)
^(^-i)S-{y-(«-»-i3+i)^}2+flft^-o
is satisfied by the two integrals
/ f^-Vl-«)T^-^-^(l-a?«)-*(i«
J
and
//-*<-
f)— >{l-(l-«)f}— A.
MISC. EXS.] HYPBRGEOMETRIC FUNCTIONS. 265
11. If
(l-ar)*-^-T'i?'(2a, 2/3, 2y, ^) = l+^ar+Cx«-|-2>a?8-|-...,•
8hew that
F{a, ft y+i, a?)/'(y-a, y-ft y+i, x)
''^+y+i^^y+i)(y+t)^^ ^ (y+i)(y+i)(y+|)^'^^ •-
(Cayley.)
12. Prove that
P^ («)=! tannir {§, «- Q.^.i («)},
where Pm(2) and Qn(^) cure the Legendre functions of the first and second kind of
order n.
13. If a function F{a^ /S, /S', y ; x^ y) be defined by the equation
F{a, ft /y, y ; x, y)° r(o)rty-„) /,*«""' (1 -«)'^~' (1 - «*)"* (I -^yT^du,
then shew that between /* and any three of its eight contiguous functions
F{a±l\ F{fi±\\ F{^±1\ F(y±l),
there exists a homogeneous linear equation, whose coefficients are polynomials in x and y.
(Leyayasseur.)
14. If y — a - 3 < 0, shew that, for values of x nearly equal to unity,
and that if y— a— ^aO, the corresponding approximate formula is
JPf a^ ^\ r(a+g) , 1
^(o,fty,^)«- r(a)r03) ^^i^^-
(Cambridge Mathematical Tripos, Part II, 1893.)
15. Shew that when \x\ < 1,
/;
^^-'(v-ar)'*-*-!^— ^(l-v)-*'Ji'
«--46'^sinair8in(p-a)ir. ^^^^°^,^^°^ i?^(a,ai,p,^X
where c denotes a point on the finite line joining the points 0, x, the initial arguments of
y-^ and of i^ are the same as that of ^, and that of (1 - v) reduces to zero at the origin.
(Pochhammer.)
CHAPTER XII.
Bessel Functions.
146. Ths Bessel coefficients.
In this chapter we shall consider a class of functions known as Bessel
functions, which present many analogies with the Legendre functions con-
sidered in Chapter X. As in the case of the Legendre functions, we shall
first introduce the functions, or rather a certain set of them, as coefficients
in an expansion.
For all finite values of z, and all finite values of t except t » 0, the
function
can be expanded by Laurent's theorem (§ 43) in a series of ascending and
descending powers of t If the coefficient of <^, where n is any positive
or negative integer, be denoted by Jn {z), we have (by § 43)
Jn{z)^^\yr---e^'^'"'^du,
the integral being taken round any simple contour in the u-plane enclosing
the point t^ = 0.
To express this quantity Jn (z) as a power-series in z, write
2t
z
Thus '^'^(')-L{i)'h~^''*'^'^''
the integral being taken round any simple contour in the ^plane enclosing
the point t^O. This can be written
-.(')-^(l)M.^'®7'— '^
146] BESSEL FUNCTIONS. 267
Now (§ 56) we have
^— . I ^"^'^Vd^sa the residue of the function ^-»--*^» e* at its pole, the origin.
If n is a positive integer, this residue is
1
if n is a negative integer, say = — «, the residue is zero when r = 0, 1, 2, . . . « — 1,
and when r ^ « it is
1
In any case, the residue is
Thus if n is a positive integer, we have
and if n is a negative integer, equal to -- 8, we have
or Jn(z)=^('-iyJ.(z).
Whether n be a positive or negative integer, the expansion can clearly
be written in the form
~ ^^ rto 2'^*-r I r (n + r + 1) •
The function Jn{z) thus defined for integral values of n is called the
Bessel coefficient of the nth order.
We shall see subsequently (§ 149) that the Bessel coefficients are a particular case of a
more extended class of functions known as Bessd fuTtctions,
Bessel coefficients were introduced by Bessei in 1824 in his " Untersuchung des Theils
der planetarischen.Stdrungen, welcher aus der Bewegung der Sonne entsteht."
In reading some of the earlier papers on the subject, it is to be remembered that the
notation has changed, what was formerly denoted by Jn {z) being now denoted by J^ (2«).
EaiompU 1. Prove that if
(l-2a^-^)H46«^ ^■*" "^^ "^^ ••• '
then will ««8in6«=^i/i(«)+^2/,(«)+^s/j(«)+....
(Cambridge Mathematical Tripos, Part I, 1896.)
268 TRANSCENDENTAL FUNCTIONS. [CHAP. XII.
For replacing the Bessel functions in the given series by their values as definite
integrals, we have
^(tR
2iriy / 2a 1 Y 46» ^
the integrals being taken round any simple contour in the u-plane enclosing the origin.
Taking a new variable t, defined by the equation
vre thus have
^.7i(z)+il^,« + ^j/,W + ...-2^.j"
<«+6« '
where the int^;ration is now to be taken in the clockwise direction round any large simple
contour in the ^plane. This expression is (§ 56) equal to minus the sum of the residues
of the function
at its poles t^ib and t=-ih ; that is, it is equal to
2t 2i
or 0" sin bzj
which is the required result
Example 2. Shew that, when n is an integer,
mss— 00
We have J^^^l^'D ^^i''^ Ji^^ ,
or 2 <"/,(* +y)= 2 <"/«(«) s rJr(y).
»— — 00 ms-oo ra— 00
Equating coefEicients of t^ on both sides of this equation, we have the required
result.
147. BeaseVa differential equation.
We have seen that, for all integer values of n, the Bessel coefficient
of order n is expressed by the formula
where C is a simple contour in the ^-plane enclosing the point ^ « 0.
147, 148]
BESSEL FUNCTIONS.
269
We shall now shew that the function Jn{z) is a solution of a certain
linear differential equation of the second order, namely,
For we find in performing the differentiations that
-7r-7ik \ t-^''^e «1-
27rt2~Jc I
n4-l ^
t "^4^
dt
= 0,
t-^.
since the function e ** f^"^ resumes its original value after the point t has
described the contour in question.
Thus Jn {z) satisfies the differential equation
d^Jniz)^ldJn(z)
dz'
dz
+ ^i-gj„(.) = o. •
This is called BesseVs equation of order n. Its properties in many respects
resemble those of Legendre's differential equation, which is also a linear
differential equation of the second order.
148. BesseVs equation as a case of the hyper geometric equation.
If c be any finite quantity, the differential equation of the hypergeometric
function
/
n
00
tc T^-^'^c z
)
— 71 — IC 2 ^ *^
)
is (§ 137)
dz^ z dz \^z z — c J
z(z — c)
If in this equation we make c tend to an infinitely large value, we obtain
270
TRANSCENDENTAL FUNCTIONS.
[chap. XII.
which is Bessel's equation of order n. Thus BeswVs eqwUion ca/n be regarded
as a limiting case of the hypergeometric equation, corresponding to the
function
/ X c \
Limit P {
n
1 .
ic 2 "^ *^ ^
0sac
\
— n — ic 2 "■ *^
Another representation of BesseVs equation as a limiting case of the
hypergeometric equation is the following.
If we change the dependent variable in Bessel's equation, by writing
e^u, the differential equation for u is easily found to be
y=
dz
I /^. 1\ du fi n*\ ^
Now if c be any quantity, the differential equation of the hypergeometric
function
P\
n
00
1
2
8
— n I — 2tc 2ic — 1
IS
dhi
d^
^ /I I 2 - 2ic\ du ^ /n^c ^ 3 ^\ u ^^
\z z — c J dz \ z 8 ) ziz — c)
If in this equation we make c tend to infinity, we obtain
cPu /I . o .\ du [ n« i\ ^
which is the above equation. Hence Bessel's equation is a limiting case
of the hypergeometric equation, being the equation for the function
e^ Limit P .
n
oc
1
2
8
— w T — ^io 2tc — 1
4
Bessel's equation is connected not merely with the general hypergeometric
equation, but with that special form of it which we have considered in con-
nexion with the Legendre functions.
148]
BESSEL FUNCTIONS.
271
For the differential equation of the associated L^eudre function (§ 129)
is (§ 138) the equation of the function
P^
-1
X
m -
2 " + 1
m
I 2
m
2
m
1-
_
2n«
or (§ 139)
/ 4n«
00
<
\
m
2
m
1
n + 1 ^ -^
m
2
— n —
m
2
The differential equation of this function is
*y / 1 1. \ c?y / m» __ n + 1 m|\ nh/
i{z^yW -4ffi^ zV d{z^)\ z^-4m,^ n z^J 2^{z^'-An^)
If in this equation we make n tend to infinity, it becomes
=0.
d{z^y^ 2^d{z
^l dy
or
r)-(->-?)|.=».
which is Bessels equation. Thus Bessels equation of order m is the same
as the equation for the function
Limit PrT (l -
2nV'
By considering Bessel's equation as a limiting case of the hypergeometric equation,
we can deduce certain solutions in the form of definite integrals.
For the differential equation of the function
'^ 00 e
ic
is satisfied by the integral
'(-:-y7.'-'(-0
^\»+i-w«
{t-zy^-hdiy
272 TRANSCENDENTAL FUNCTIONS. [CHAP. XII.
if C is a ooDtour such that after describing C the integrand returns to its initial value.
When c becomes infinite, this expression reduces to
sf^tr*^ j <-»-i(<-«)-»-ie«'cfc,
which accordingly satisfies BesseFs equation if C be a contour of the kind described ; C can
for instance be a figure-of-eight contour encircling the points t=^0 and t=:z.
In fact, if we write
we have
=0.
Other solutions can be found by changing the signs of n and t.
Example. Shew that Bessel's differential equation is the limiting case of the equation
of the hypergeometric function
00 c*
P\ in i(c-n) ^
when c tends to infinity.
149. The general solution of BesseVs equation by Bessel functions whose
order is not necessaHly an integer.
We now proceed, in the same way as in § 116, to extend our definition of
the function «/n(^) to the general case in which n is not an integer.
It appears from the proof given in § 147 that, whatever n may be, the
differential equation
is satisfied by an integral of the form
y=z'' r'*-^ e *^ dt,
J c
provided the path of integration C is a contour on the ^-plane, so chosen that
the function
e «r^-i
resumes its initial value after describing C.
149] BESSEL FUNCTIONS. 278
Now when the real part of Ms a very large negative number, the
function
is infinitesimal. Hence y will be a solution of the differential equation,
provided the contour G begins and ends with values of t whose real part
is infinitely large and negative.
Let therefore a contour C be taken which begins at the negative end
of the real axis, and after proceeding close to the real axis to the neighbour-
hood of the origin makes a circuit of the origin and returns, close to the real
axis, to the negative end of the real axis again. The integral y taken round
this contour satisfies BesseVs differential equation.
We shall now shew that this solution y can be expressed in the form of
a series of powers of z.
Suppose as usual that by tr^^^ is understood that branch of the function
ir^-^ which when continued (§ 41) to the point ^ = 1 by a straight path,
arrives at the point ^ = 1 with the value unity.
Then we have
y = £«| r^>e^6 **<fo
00 / ly-.sr-fn r
r=o ^.r\ Jc
But (§ 100) we have
But when n is an integer, we have (§ 146)
^"^^^^ ^to2^^r! r(n + r + l)*
Comparing these results, we have, when n is an integer,
•^" <^> = 2^- (l)7c'"^'*'"^'"'
where C is the contour already described.
Now we have seen that the right-hand side of this equation has a meaning
and satisfies BesseFs differential equation for all values of z and all values of n ;
whereas, up to the present, Jn{z) has been defined only for integral values
w. A. 18
274 TRANSCENDENTAL FUNCTIONS. [CHAP. XII.
of n. We shall take this opportunity of extending the definition of Jn (z), in
the following way.
For aU valuea of n and of z, the function
h 'Silr''"'"'
CO (,l)r^gir+n
rro2~+*-r! r(w + r-!-l)
or
will he denoted by Jn (z). In the integral, tr^"^ is to have the value which
becomes unity when the variable t travels in a straight line to the point ^ = 1 :
and is a contour which encircles the point ^ = and begins and ends at
the negative end of the real axis in the ^plane. The function Jn(z) thus
defined is called the Bessel function of z, of the first kind and of the order n ;
it satisfies BesseVs differential equation of order n.
Since Bessel's differential equation is unaltered by the change of n into
— n, we see that J^n(z) is also a solution of the equation ; and therefore the
general solution of BesseVs equation is of the form
aJn (z) + bJ^n (z\
where a and b are arbitrary constants, except in the case in which Jn (z) and
«/Ln (z) are not independent functions ; this exceptional case happens when n
is an integer, for then, as we have already seen, we have the relation
Jn (Z) = (- irJ-n (*).
A second solution of BesseFs equation in the case when n is an integer
will be given later.
160. The recurrence-formulae for the Bessel functions.
As the Bessel functions, like the Legendre functions, are members of
the general class of hypergeometric functions, it is to be expected that
recurrence-formulae will exist between them, corresponding to the relations
between contiguous hypergeometric functions (§ 145).
We shall now establish these recurrence-relations ; the proof given does
not assume the order n to be an integer, and consequently the formulae are
valid for all values of n, real or complex.
Let C be the contour described in the last article, which begins and ends
at the negative end of the real axis in the ^plane, and encircles the point
t = 0.
Then since the function
e ^ir^
¥mmt^mBsm
150, 151] BESSBL FUNCTIONS. 275
is infinitesimal at the extremities of this contour, we have the equation
2n
or */n-i (^) + */n+i (-8^) = — t/n (-2) (A),
z
which is the first of the recurrence-formulae.
Next we have, by differentiation,
or
''-^.- -,■''<.')- J-^M (B).
From (A) and (B) it is easy to derive other recurrence-formulae, e.g.
^-i^ = \[j^,{z)-j^^,{z)] (C),
and ^!^) = j^,(,)_^/,(,) (D).
Example 1. Shew that
16'^^=^.-4(«)-4^«-,W+6/»(«)-4y,^,(*)+J,^,W.
Example 2. Shew that
161. Relation between two Bessel fimctions whose orders differ by an
integer.
The various recurrence-formulae found in the last article can* however be
easily deduced from a single equation, which connects any two Bessel
functions whose orders differ by an integer, namely
n+r
w^/ ly _*;
(-ly
z"^^ "" ^ (zdzy
where n is any number (real or complex) and r is any positive integer.
18—2
276
TRANSCEKDENTAL FUNCTIONS.
To establish this result, we have, by § 149»
dr {Jn{z)
d{s^y\ z""
dr I f
d{j^y27n.2''Jc
e *tdt
dt
[chap. XII.
(- 2/ -2^+'"
•^ n+r V-*^)*
which is the equation required.
The recurrence- formulae can be derived without difficulty from this resultw
Thus, equation (B) of the last article is obtained by taking r = 1 in this
equation : and equation (A) of the last article may be derived in the following
way.
Taking r= 1 and r = 2 successively in the formula just proved, we can
express the first and second derivates of Jn(z) in lerms o{ Jn{z), Jn+i{z) and
Jn+2(z), in the form
dJn (z) _ n J. ,. J , .
— ^^ =- - (1 - n)Jn (Z) y-^n+i W + «/n+s(4
Substituting these values in BesseVs equation
we have
Changing n to (n — 1) in this result, we have
J,+,(z)-^/,(*)+J._,(^) = 0,
z
which is the formula (A) of the last ai*ticle.
The other recurrence-formulae can be derived in a similar way.
152, 153] BESSEL rUNCTIONS. 277
162. The roots of Besael fwncHons,
The relation established in the preceding article enables us to deduce the
interesting theorem that between any two consecutive recU roots of Jn {z) there
lies one and only one root of Jn+i (^)*.
For since Jn(z) satisfies BesseFs equation, it follows that the function
y = Jir^ Jn (z) satisfies the differential equation
From this equation it is evident that if f be a value of z (real and not
zero) for which ^ is zero, then the signs of -~ and y must be unlike at the
point 2r = f . Now let -^ = ^i and ^ = f, be two consecutive roots of the
function -^. It is clear from the differential equation that neither y nor -~
can be zero at either of these points. Then the function -^ -~ has a
different sign just before reaching 2^ = fa to that which it has just after
leaving -? = fi; and hence it follows that the function y -^ has a different
sign just before reaching z= ^^to that which it has just after leaving z^ fj.
The function y must therefore have an odd number of roots between the
points 2r = f 1 and z = fj.
But from Rollers Theorem it follows that y cannot be zero more than
once in this interval : so y must have one and only one zero between the
points 2^ = f 1 and 2^ = fa • and therefore the zeros of y and of -p occur
alternately.
Thus, between any two consecutive roots of the function z~^ Jn (z) there
lies one and only one root of the function -j- {z~^ Jn (z)] or - z~^ J^^i (z): which
establishes the theorem.
163. Expression of the Bessel coefficients as trigonometric integrals.
We shall next obtain a form for the Bessel coefficients (ie. the Bessel
functions for which the order n is an integer), which in some respects
corresponds to the Laplacian integrals obtained in ^ 119 and 132 for the
Legendre functions.
* The proof here given is dae to Oegenbaaer, MonaUhefU fiir Math, ym. (1897).
278 TRANSCENDENTAL FDNCTIONS. [CHAP, XII.
If Id the equation
we write t = e^, we have
n»-ao
Changing i to — i in this equation, we have
»« -00
Adding and subtracting these results, we have
00
COS {z sin ^) = 2 J^ (z) cos 7uf>,
n« -
sin (z sin <^) = S Jn (z) sin n^.
n«-flo
Since J^ (z) = (— l)'*«7Ln (A these equations give
cos(* sin <l>)^Jo (z) + 2J, (^r) cos 2^ + 2J4 (2r)c08 4<^ + ...,
sin {z sin ^) = 2Ji (^) sin ^ + 2/, (5) sin 3^ + ... .
As these are Fourier series, we have (§ 82)
J^ (-?) = - I cos nO cos (2fsin 0) dO, (n even),
1 r*
= - I cos ntf cos (z sin tf ) dtf , (n odd),
1 f*
J^ (fr) = - I sin 72^ ain (2? sin 0) dd, (n odd),
1 r*
= ~ I sin nd sin (z sin ^) dtf , (n even).
TT Jo
Since
cos (ntf — e sin tf ) = cos nO cos (^ sin 0) + sin ntf sin (z sin tf),
we have in all cases when n is an integer
1 f*
Jn (z)=i— I COS {nd — J? sin 0) d0,
TT J Q
the formula required.
Example, To shew that for all values of n, real or oomplez, tbe integral
1 fw
y—- I ooe (n$ - $ mn $) dS
ir J
154] BESSEL FUNCTIONS. 279
satisfies the differential equation
^ ^ dy . f-, w'N sin nn /I n\
which reduces to BessePs equation when n is an integer.
1 /■»
Forif y = - I cos(n^-28ind)eW,
wehave 7^^~ I sindsin (?i^-«sin^)cW,
so
cPv 1 r*
-^= I sin*^cos(n^-«8in^)cW,
p
y+^ = - / cos*dcos(7id-«sind)cW,
and " "J"^ ^y^~ \ sin(nd— 2Sind)cW — I pCos(nd-«smd)cW.
Now integrating by parts, we have
-I — — sin(n^-2sin^*e^«— sinnTT-l — I — (n-;^cos^)cos(w^-«sind)eW,
vjo z nz vjo z ^
and therefore
1 t A -- ^
«= — sinn7r--«- / cos(n^-«sind).rf(n^-«sind)
nz «"«• y ^»o
1 . n .
sin?tir /l _»\
which is the required result.
164. Extension of the integral-formula to the case in which n is not a/n
integer.
We shall now shew how the result
1 f »
(xr) = - I cos {nd — zsmff) dO
must be generalised in order to meet the case in which n is not an integer,
ie. the case of the Bessel functions^ as opposed to the Bessel coefficients.
280 TRANSCENDENTAL FUNCTIONS. [CHAP. XII,
Suppose that the real part of z is positive. Write ^ = ^ ru in the formula
we thus have
where u"*^^ has that value which becomes unity when the variable u travels
by a rectilinear path to the point w = 1. Since values of t whose real part is
large and negative correspond to values of u whose real part is large and
negative, we see that the path in the u-plane, along which this integral is to
be taken, is still a path leading from u = — x round the point u = and
returning to w = — oo .
Let this contour be chosen so as to consist of
(a) a straight line parallel to, and below, but indeBnitely close to, the
real axis from it = — oo to w = - 1 ;
()8) a circle I of radius unity described round the origin ;
(7) a straight line parallel to, and above, but indefinitely close to, the
real axis from u = — 1 tot4 = — 00.
Thus
'^»<^> = 2^/_">-^'^^""-^'^« + 2^/,«-""^
1 /•-• ?U-^)
ZTrt j-i
where u~*"* has in the first integral the value «<*■♦■»)*» at u = — 1, and in the
third integral has the value 6~<'^*^*' at u = — 1. Hence, writing w = — t in
the first and third integrals, and u = e^ in the second integral, we have
1 r* ^(»+i)t» /•• -^-f+-\
dt
where, in the last two integrals, t"**~* has the value 1 at the point t«l.
Writing t = «•, we have
^ sin(n + l)7r r^_^.,^„h#^^
TT Jo
^trjo ' TT Jo
154] BES8EL FUNCTIONS. 281
or Jn(z) = - ['coaizsine-neyde-''^^^'^ I e-»»-"inh«d^ (1).
This formula is valid when the real part of £: is positive. When the real
part of z is negative, a similar procedure leads to the result
J^ (z) = ^— j r cos (z sin + n^) d0 - sin nir j e-«*+'»inh<> ^^1 (2).
When n is an integer, the formula (1) gives
1 f*
J„(^) = - j cos (w^ — 2: sin ^) d^,
'jr Jo
when the real part of -^ is positive ; and the formula (2) gives
J„ {z) = tllZ! r cos {ne + z sin 0) d0,
or, since /« i^) = (- 1 )** J-^ {z\
1 f""
t/n (-2^) = - cos (nd — 2^ sin 0) d0,
TT Jo
when the real part of z is negative.
Thus in either case when n is an integer, we have again the result of the
last article, namely the formula
'„(^)=1 rcos(/i^-2rsin^)(i^ (3).
The equation (3) was kDown to Beasel. Equation (1) is due to Schlafli, Math. Ann. m.
<1871) ; equation (2) was first given by Sonine, Math. Ann. xvl (1880).
The trigonometric integral-formula for J^ (z) may be regarded as corresponding to the
Laplacian definite integrals for the Legendre functions. For we have seen that the
Bessel function J„^ (z) satisfies the differential equation of the function
Limit P^
11—00
or
or
But the Laplacian integral shews that this quantity is a multiple of
^* I'ob "£"' "^ (0 " £')*" i}*««*J*=°"'^ '^
Limit I fl-i — cos<^] ooHfmf>d<l>f
I ^OM^GOBin<f>cUl>j
the similarity of which to the above result (3) will be observed.
282 TRANSCENDENTAL FUNCTIONS. [CHAP. XII.
166. A second expression of Jn {s) as a definite integral whose path of
integration is real.
Another definite-integral formula, which is valid for all values of z and a
certain range of values of n, can be obtained in the following way.
The function Jn (z) is expressed for all values of n and z by the series
I (-l )r^+«r
rro2'»-^r!r(n+r+l)*
Since (§ 95) we have
this can be written in the form
(-l)-z"^r(r + l)
Jniz)= S
■0 2»r(n + r+l).2r!rQ) *
Now by § 107 we have
provided the real parts of (r + ^ ) ai^d (n -f = ) are positive.
Thus if the real part of ( '^ + «) ^ positive, we have
•^nW«-— TXr— 7 n 2 ^ — ^ cos^'V^sin'^i^d^.
2«ryjr(w + ijr=o ^r\ Jo
Tj ^ / .X ^ (-l)''^cos*'^
But cos (£^ cos <p) = 5) ^— -. , ^ .
^ ^' r^^Q 2r!
Thus we have .
J (z) =s J, I J .1 I COS (2: COS 6) sin** ^ (i<5.
2-rQ)r(n+y'o
This formula is true for all values of z, and for all values of n whose real
part is greater than - ^ .
Example 1. Shew that
P^ (cos 6) « jT^^fYj j^ e''<^» Jo (^ ffli^ ^) ^ dx.
80
155, 156] BESSEL FUNCTIONS. 283
For we have
fry
Jo ir J J
= r(n+l)P»(co8^),
which establishes the result.
Example 2. Shew that
Pn"» (cos ^)= -7—4-7 fx f e''^9J„,(xamB)x^dx.
r(n— m+i;y o
(Cambridge Mathematical Tripos, Part 11^ 1893.)
166. Hankers definite-integral solution of BesseVs differential equation.
If in the result of the last article we write
t = cos (f>,
we obtain the result
2"r(i)r(n-.y^-i
It will now be shewn that this integral is a member of a very general
class of definite integrals which satisfy Bessel's differential equation, namely,
integrals of the form
y=^n f e^ (t^ ^ l)n'h dt,
Jo
where C may be any one of a number of contours in the ^-plane. The
importance of solutions of this type was first shewn by Hankel*.
To shew that integrals of this class satisfy BessePs equation, we form the
first and second derivates of the expression y, and find that
= -z^' ( {^e«'((»- !)"+»- (2n + l)He^ («•- I)""*}
J c
dt
* Math. Ann. i.
284 TRANSCENDENTAL FUNCTIONS. [CHAP. XII.
From this it is clear that Bessels equation will be satisfied by the integral
J c
provided is a closed contour such that the integrand resumes its initial
value after making a circuit of C
The similarity of this result to the general theorem of § 142 is very
apparent.
167. Expression of Jn (z), for all vaiues of n and z, by an integral of
HankeVs type.
We shall now shew how the particular solution Jn {z) of Bessel's equation
can be expressed by an integral of HankeFs type. Consider the contour
formed by a figure-of-eight in the ^plane, enclosing the point ^ = + 1 in one
loop and the point ^ = — 1 in the other, so that a description of the contour
in the positive sense involves a turn in the positive direction round the point
< = + 1 and a turn in the negative direction round the point ^ = — 1. After
turning round the point < = + 1 in the positive sense, the integrand resumes
its original value multiplied by ^»-*>*'*, as can be seen by writing it in the
form
««+(n-i)log«-l)+(i»-4)log(e+l).
^ »
and after turning round ^ = - 1 in the negative sense, it is further multi-
plied by
-(i»-i)fcrt
Hence after describing the whole contour, the integrand resumes its
original value.
rd+.-i-)
Thus y^i^\ e**(e«-i)»-*(ie
is a solution of the differential equation, valid for all values of z and of n ;
the symbol (1 +, — 1 — ) placed at the upper limit of the integral indicating
that the path of integration consists of a positive revolution round 1 and a
negative revolution round — 1
In this equation we shall suppose as usual that ^ has the value which
reduces to I when z travels by a straight path to the point z^\, and we shall
suppose (^" — I)*"* to have initially the value which reduces to «-<*-*)'» when
t travels by a straight path to the point ^ = 0.
To find the relation between this quantity y and the particular solution
Jn {z) of Bessel's equation, we expand y in the form
157] BESSEL FUNCTIONS. 285
To evaluate the iutegrals which occur in this series, write
F{r,n)^j
(1+.-1-)
m+,-1-)
Then F(r, n+ l) = j (r+«-r)(<«- !)»-*(£«
a+.-i-) ^H-i
-I
/•(i+.-i-)(««_l)«.+i(r + l)^- „, .
Thus we have F{r, n) = - ^ ^"T" jP(r, n + 1).
This result enables us to reduce the evaluation of F(r, n) to the evalua-
tion of F(r, n + 1), and thus to the evaluation of F(r, n + k), where A is a
positive integer so chosen that the real part of (n + A:) is greater than — ^ •
We have therefore to evaluate the integral
F(r, n)=|'
where we may now suppose that the real part of n is greater than - ^ . The
contour can be supposed to start at the point t = 0, where (^'— I)**"* has the
value «-<'»-*>»», then to proceed to the neighbourhood of the point ^ = 1 along
the real axis, then to make a positive turn in a small circle round t = 1, then
to return along the real axis to the point ^ = 0, where (^*— 1)**"* has now
the value e<»-*)'^, then to proceed along the real axis to the neighbourhood
of the point ^ = — 1, then to make a negative turn in a small circle round
^ = — 1, and lastly to return along the real axis to the point t = 0, where
(^— l)**"* has now the value e"^""*^**. Since the real part of n is greater
than — 2 , the integrals round the small circles at ^ = 1 and ^ = — 1 are
infioitesimal, and we therefore have
^0 Jo
dt
JO Jo
where in each of these integrals the quantity (1 — <■)*"* is now supposed to
286 TRANSCENDENTAL FUNCTIONS. [CHAP. XII.
have the value unity at ^ = 0. Writing — ^ for ^ in the two last integrals,
we have
F(r, n)= - {e^"-*)** - «-<»-*)'<} {1 - (- 1)^-1} JV(1 - t*)^-^dt
= — 2% cos* -x-sin(n— sJtt/ v* <^"*^ (1 — i/)*-* dv, where v = <",
<by § 105) = - 2i C08' ^ sin (n - 1) ^fi (^ , n + ^)
(by §106) =-2ico8'^sin(n-l)^ ) r \^' '
r^n+2 + l)
This result has now been proved to hold so long as the real part of n is
greater than — = : and in virtue of the formula
F(r. n) g^^^ Fir.n + 1),
we see that it holds universally.
Thus we have F(r, n) = 0, when r is odd ; and it is therefore sufficient to
take r even. Let r = 2s. Then the formula becomes
^•(2,, n) = 2i sin (» + 1) ^ \^/; \ , V .
But (§ 97) we have
r(„+l)rQ-„) —
TT
sm
Therefore 2^V^/'« + l^
/'(2», n)=— ^^ -_i U .
r(i-n)r(n + , + l)
and so y = 2 ^^ — a-, M <
'"> 2«! r (i-n)r(n + «+l)
or
,^ » (-l)^+» ^^"^^ (§)
^ .=« 2-«! r(i-n)r(» + ,+ i)-
But
"^ ' ,ro2-+»«!r(n + «+l)*
158] B£SSBL FUNCTIONS. 287
Therefore J„(.)=_^-^,.
(ly
This formula gives the required expression of J^ {z\ It is valid for all
values of n and of z\ but when n is of the form (^ + 2)» where A is a
positive integer, the factor T (o — w ) becomes infinite and the integral
/
becomes zero (since the integrand is now regular at all points within the
contour), so that for this exceptional case the formula is indeterminate.
Example, Deduce the formula
from the result of this article.
«
168. Bessel functions as a limiting case of Legendre functions.
We have already (§148) shewn that Bessel's differential equation of order
771 is the same as the differential equation of the associated Legendre
functions
We shall now express this connexion more precisely, by establishing the
formula
^„ (.)= Limit n-.P,«.(l-^.).
For taking the expression of the associated Legendre function by a definite
integral (§ 131), we have
~mpm(-t _ ^ (n-\-m){n + m-\) (n-TO + 1)^" / ^N**
" " V 2»»y" (2to - 1) (2m - 3) l.-n-.n"" \ 4m*}
/„'{^-£« + '^*(-| + £')P«^""^'''^'
and as n becomes infinitely great, the right-hand side of this equation tends
to the limiting value
7^5 rr-To ^r = / ( 1 + - COS ^) sin"*Ad<f>,
(2m — l)(2m — 3)...l .TrJo \ n v ^ ^'
288 TRANSCENDENTAL FUNCTIONS. [CHAP. XU.
— o)... 1 . TT Jo
(27?i-l)(2m-3)
or (§165) Jn.{zy>
which establishes the result stated ; it is due to Heine*.
169. Bessel functions whose order is half an odd integer.
The result of § 157 suggests that when the order n of a Bessel function
Jn (z) is a number of the form A? + 5 , where A? is a positive integer, certain
exceptional circumstances arisef in connexion with the function. In this
case it is in fact possible to express the Bessel function
•^*^^^^"2*+*r()fc + ?) I 2(2A: + 3)'*'2.4.(2A; + 3)(2A + 5) ••*)
in terms of well-known elementary functions.
For by § 161 we have, if A; be a positive integer,
/„«-(-2)..«4^|-'.W}.
But the series-expansion of the function J^ {z) is
2\i .
sin^r.
. , , 2M f, z* z* ) / 2 \
Therefore J,^i {z) -^ ^ ^^,^ (-^J ,
which is the required expression of the function Jj^ {z) in terms of more
elementary functions.
The student will without difficulty be able to prove that a second solution of Bessel's
differential equation in this case is
* Heine's definition of the associated Legendre function is somewhat different from that
which has since become general and which is adopted in this book : this leads to differences of
statement in many other formulae, such as that of this article.
t The student who is familiar with the theory of linear differential equations will observe that
in this case, and also in the other exceptional case of n an integer, the difference of the roots of
the '* indicial equation *' of BessePs equation is an integer.
159, 160] BESSBL FUNCTIONS. 28&
Example. Shew that the solution of the equation
IS y-« 4 2 Cp{*^-m-* (2apl*)+»J'« + i(2a^)},
p»0
where the quantities Cp are arbitrary constants, and o^y a|, ... o^mi &x^ ^^ roots of the
equation
o***^— i (LommeL)
160. Expression of Jn (z) in a form which furnishes an approximate value
to Jn {z)for large real positive volumes of z.
We now proceed to form an integral which will be found to play the
same part in the theory of the function J^ {z) as the integral of § 104 plays
in the theory of the function T {z). We shall suppose 5 to be real and
positive. Then, by § 155, we have, for all positive values of n,
J^ (z) SB / cos {z cos <f>) siD^<f> cUf>,
Writing cos <^ = a?, this becomes
Jn(z)=^ J 11 I (1 - a^y^ COS zxdx,
or Jn(-^) = Real part of /^ . /^(l - x')'^e^dx.
In order to transform this integral, we take in the plane of a complex
variable t a contour OPQBCOy formed in the following way. is the origin
(^ = 0) ; P is the point ^ = 1 — />, where /> is a small quantity, and OP is the
part of the real axis between and P. Q is the point ^ = 1 + 1/), and PQ is a
quadrant of a circle which has its centre at the point ^ = 1. B ia the point
t^l -hik, where A; is a large positive quantity, and QB is the line (parallel to
the imaginary axis in the ^plane) joining Q and B, C is the point t = ik, and
BC is the line (pcurallel to the real axis) joining B and C, Lastly, CO is the
part of the imaginary axis between C and Q. Then the function
is regular at all points of the ^plane in the interior of the contour OPQBCO ;
and therefore the integral
taken round this contour, is zero.
w. A. 19
290 TRANSCENDENTAL FUNCTIONS. [CHAP. XIL
We can write this relation in the form
f +f +f +f +f =0.
J OP J PQ JOB JbC J CO
Now the part of the integral due to PQ tends to zero with p, and the part
due to BC tends to zero as k becomes infinitely great, while the part due to
CO is purely imaginary. Thus we have
Real part of / = — Real part of I ,
J OP J QB
and so Jn W= Real part of , ^ ,i f (1 -fy*^^cU.
In this integral write
%u
z
80 that u varies between the limits and oo when t describes the line QB;
and (i_^)* = /f(?y«*(l + gy.
and therefore f =2"-*w {"*'""^r-»-*rr^M-* (l + g)*~*dM.
Thus we have
or j; («) =
r(n+i).(27r^)»
'-{-(-i)i}/>»-i(>4r-('-ir}*''
-"{'-(«4)f}/:— {(>-ir-('-£n<<'«j
This is the integral-expression required. It is easily seen to furnish an
approximate value of J^ (z) for large positive values of ^ ; for as ;? becomes
indefinitely large, the two integrals in the expression tend respectively to the
limits 2r (n + ^ j and zero ; and therefore the function Jn (z) approximates
for large positive values of z to the value
(l;)*^|'-KI)l}-
160] BESSEL FUNCTIONS. 291
The evaluation of «/» (z) when z is large will be considered in fuller detail
in the next article.
The result of this article can also be obtained in the following quite different manner,
which connects it more closely with the general theory. We have seen in § 148 that Bessel's
differential equation is a limiting case of the general hypergeometric equation, represented
by the function
00 e
n i-2tc 2w?-l
Since the differential equation of the P-fimction
(0 00 c
a p y Z
is (§ 142) satisfied by the integral
taken between suitable limits, we see that Bessel's equation is satisfied by the expression
Limit «<•«-* (<»-» (l-^y •-»+«• («-0*'*<^,
or ««•«-* [ r-J «-«< {z - <)•"* dt,
or (putting < «= - ivz)
«<•«-» jif''-hs^'k{z+%vz)*-he-^zdv,
or «*V» / (v+tv«)*-*e->"cfo.
The limits of the integral can be taken to be and oo , since these satisfy the conditions
for the limits found in § 142 ; and hence it follows that
is a solution of Beesel's equation.
Similarly the quantity
is a solution of Bessel's equation.
The solution J^ {z) must therefore be of the form
J^(z)»A^i^ r {v+iv^'ie'^dv'¥Be'*'s^ riv-iv^'^'ie'^dv,
19—2
292 TRANSCENDENTAL FUNCTIONS. [CHAP. XII.
where A and B are constants independent of i. This is substantially the form given above,
but the determination of the constants A and B \b a, matter of some difficulty, for which
the student is referred to a memoir by Schafheitlin, CrelU^s Journal^ oxiv. p. 39.
Example, Shew (by making the substitution u^2zcot<t>m the integral found above^
or otherwise) that
ir
J^{s)^ ?^^^^?!^. fV««~**oos«-*<^oosec*»+i<^cos{«-.(n-i)^}c&^.
r(n+i).v*Jo
161. The Asymptotic Expansion of the Bessel functions.
The Bessel functions can for large values of the argument be represented
by asymptotic expansions. We shall here consider only the asymptotic
expansion of Jn{^) for positive real values of z; this was discovered by
Poisson (for n = 0) and Jacobi (for general integer values of n). The theorem
has been considered for complex values of z by EUmkel * and several subse-
quent writers.
We shall derive the asymptotic expansion from the integral-expression
Jn(z) =
(iwz)^ r (n + 1)
■-('-(- 1)1} /:--'i('4r-(' -sn^»
found in the last article.
It is first necessary to find the asymptotic expansion of the integral
which we shall denote by the symbol /.
Now we have
iu
^.jt2Li^)r(g..)-(>,.^.^.
* Math, Ann, i.
161] BESSEL FUNCTIONS. 293
Therefore
tu
^ Mt-i)^..(t-.) jv^.._/;(g-.)-(i^.)^.^,
or
where
Now as z becomes infinitely large, n having any definite finite integer
value, the remainder-term R^^ tends to the limit
It follows from this that
Limit z^R^ = 0,
and therefore the series
r(ifc+ 1) jlH- i^(A^ + r)(^-Hr-l)...(fc-r + l)0|
is the asymptotic expansion of the function
r e^u^fl + ^^j'du (k>0).
Substituting this result in the expression already found for «/,(-?), we
see that
rU^)
(2,r^)* r(n + i)
^r(" + 2)2tf + i. W (fe^l
^- i f ■ l\-^)f^ (""^•^'')("~^+'')-("~^-^5)i'-^.-,-.(-t)r |
COS
294 TRANSCENDENTAL FUNCTIONS. [CHAP. XIL
is tbe asymptotic expansion of the Bessel function «/n(^) for large positive
values of z.
Even when z is not very large, the value of c/» {z) can be computed with great accuracy
from this formula. Thus for all values of z greater than 8, the first three terms of this
asymptotic expansion give the value of Jq (z) and Ji (z) correct to six places of decimals.
162. The second solution ofBesseVa equation when the order is an integer.
We have seen in § 149 that when the order n of Bessel's differential
equation is not an integer, the general solution of tbe equation is
where a and fi are arbitrary constants.
When however n is an integer, we have seen that
and consequently the two solutions Jn (*) and «/_n (z) are not really distinct.
We therefore require in this case to find another particular solution of the
differential equation, distinct from Jn(^)t in order to have the general
solution.
To obtain this second solution, we write
y = uJn (z),
where u is a new dependent variable, in Bessel's equation
Remembering that Jn(^) is a solution of Bessel's equation, the differ-
ential equation for u becomes
162] BESSEL FUNCTIONS. 296
dj^ a dz . 1 rt
or -_ + 2-y— — + -=rO.
dz
Integrating this equation, we have
UtUt
log ;j- + 2 log Jn {z) -h log z = constant,
dvt 6 . « •
or -=- = . - . .. , where 6 is a constant,
dz z [Jn {z)Y
where a and 6 are arbitrary constants.
The complete solution of Bessel's equation can therefore be written in
the form
To find the nature of the solution thus obtained, we observe that in the
vicinity of the point ^ = the integrand
is of the form
^~**~* (constant + powers of f)~*,
which when n is a positive integer can be expanded as a Laurent series in
the form
The function
l'tr'[Jn{t)]''dt
has therefore the form
where the quantities d^^, ^^-m+s* ••• ^^^ definite constants.
It thus appears that the complete solution of fiessel's equation can be
written in the form
y = ilJn(^) + 5{/»(^)log^ + t;},
where v is the result obtained by multipl3nng together J^{z) and a Laurent
series of the form
296 TRANSCENDENTAL FUNCTIONS. [CHAP. XIL
where the quantities (£-an> <2-sn+af ••• ^^^ definite constants, and A and B are
arbitrary constants. The expansion of Jn (^) being known, we see that the
product V has the form
jt^ X a power-series in ii* ;
and thus a second solution of BesseVs differential eqiwiion, in the case in
which n is an integer, can be taken of the form
Jn (z) log 2r + 2r« (Oo + OiZ* + Oi-e* + a,;^ -h ...),
where the quantities o^, Oi, a,, ... are definite constants. These quantities a
are not however all of them strictly speaking definite, since by adding a
multiple o{ Jn{z) (which will leave the expression still a solution of Bessel's
equation), it is possible to change all the quantities a after On-i.
This solution will be denoted by Kn{z)*.
The coefficients Oq, Oi, a,, ... may theoretically be determined by substi-
tuting this expansion in the differential equation, and equating to zero the
coefficients of successive powers of z, A better method is however the
followingf.
We have seen that when n is a positive integer, J.^(z) reduces to
(— l)^Jn{z); in fact, if in the equation
T ( \ l^X'"'^^ T (-1)^ /^\^
J-(«-.,W=^2^ p:or(-n + 6+p-hl)r(2> + l)V2;
we suppose the quantity € to tend to zero, all the terms of the series vanish
as far as p = n, since r(— w-hp + l) is for these terms infinite. Changing
the meaning of the index of summation /> in the other terms, we have
./_(„_, w W p.«r(-n+e+;)+i)r(p+i)Uy
^ ' \v ,tor(€+p + i)r(n+p + i)V2y '
and when e = 0, the first of these partial series is zero and the second is
(-l)»J„(e).
Since the quantity
(-1)»J_,^,(^)-J,„_,(^)
vanishes with e, we can take as a second solution of BesseVs equation the
limiting value of the quotient
* In referring to memoirs it most be borne in mind that different writers have taken different
definitions of the Besael fonctions of the second kind,
t Dae to Hankel, Math, Ann, x. p. 470 (1869).
162] BESSEL FUKCnONS. 297
Substituting the above values for the Bessel functions, this becomes
\2) ^toT(p + l)€T{-n + e + p+l)\2) '*'V2/'pto^ ^ e \2) '
where /(€) represents the expression
^^^^"r(n+p + i)r(p + 6+i)(2J "r(n+p-€+i)r(p+i)V2J '
The limiting value of /(€)/e, as € tends to zero, is
^ 21og^ ^ ^<^ + ^>
1 r(«+p+i)
r(p+i){r(n+p +!)}••
Also, since
we have limit ,r(_„^^,^^^i) = (- l)"^' r (« - p).
Consequently we obtain, as a second particular solution of Bessel's
equation, the expression*
\2; ptor(p+i)V2; ^{2) pZ,,r(n+p+i)T{p+i)\2)
z r(n+p + l) F(p + 1)]
r"'*2 r(n + p + i) r(p + i)j*
The coefficient of log« in this expression is 2J,(«). So, dividing the
expression by 2, we have tbe second solution in the form '
j.(«)i»g.-^;i;E^(i)''-^.wiog2
+
(iXi. (-i)p cy^l ^(n+p+ i) r(p+i )]
W pr«r(n+p + l)r(j) + l)V2/ 2 1 T(n+p + l) r(p + l)j
It is convenient to add to this e^cpression a term
/,(.){log2+M)|.
* This 18 Bankers seoond solution Y^ (z). It is really
dn ^ ' dn '
298 TRANSCENDENTAL FUNCTIONS. [CHAP. XU.
which is itself a solution of Bessel's equation ; so the second solution now
takes the form
^n(.)iog.-2^2— |j-^y
^\i^S\ (- \y /^\^ [F (n -h p + 1) r (p +1) 2r (i) ]
2Wptor(n+p+i)r(i)+i)V2>' lr(^+i) + iK r(p+i) r(i) r
This is the solution K^ {z) which we take as our standard.
Since, when r is a positive integer, we have
r(r + l) F(l)_ 11 1
r(r+l) r(l) '■^2'^3'^"'^r'
we can write K^{z) in the form
_|(|ri^z^j,+ui+...+i+i+|+...+ i_i(|r.
2\2/ ^«o (»H-l>)lpl I 2 3 p 2 7i+pJ V2/
When n is an integer, the two independent solutions of Bessel's differ-
ential equation are J^ {z) and K^ (z\
Example 1. Shew that the function K^ (z) satisfies the recurrence-formulae
These are the same as the recurrence-formulae satisfied by J^ («).
Example 2. When the real part of 2 is positive, shew that the expression
r8in(isin<^-n<^)ci(^- j e-*«toh«{e««+(-l)*e-«^ dS
is a second solution of Bessel's differential equation of integer order n.
(SchlaflL)
Example 3. Shew that the expression
yologi+2(y,-^4+iJ5-...)
is a second solution of the Bessel equation of order zero.
163] BESSEL FUNCTIONS. 299
163. Nexmianns expansion ; determination of the coefficients.
We shall now consider* the expansion of an arbitrary function f(z),
regular at the origin, in a series of Bessel functions, in the form
f(z) = OqJo (z) + a^Ji (z) + aa/2 {^) + " •»
where the coefficients a©, «!, ««> ••• are independent o{ z. ,
Suppose first that such an expansion is possible, and let us try to
determine the coefficients, by expanding both sides of the equation as
power-series in z and equating coefficients of the several powers of z. Since
/(^)=/(0)+2(|)/'(0) + |(|)V"(0)+|^(|)V"'(0) + ...
and «^»W = ^(|) {l-i!(„Vi)(|) +2!(n + lKn + 2)(l) --}'
we have on comparing coefficients the equalities
/(O) = «..
2/' (0) = a„
2"/" (0) = - 2a. + Oa, etc.,
from \rhich without difficulty we find
a. = /(O).
«„ = 2 1/(0) + JV" (0) + "'^"4;"^'^ /'^ (0) + . . . + 2»-»/ <«) (0)| (n even).
«„ = 2|n/'(0)+"-(^^V"(0)
+ »(»'-l')("'-3') ^,T) (0) + . . . + 2»-'/ <») (0)1 (n odd).
These coefficients take a simpler form, if we introduce functions Oi {z),
Ot (e), 0, (z), ..., defined by the formulae
^,. » n(n*-V) n (n» - !•) (n» - 3") 2»-in! . ,,,
for then it is easily seen that On is twice the residue of the function On(t)f(t)
* 0. Nemnann, Theorie der Bet$eV$ehen Functionen. The exposition here given followB
Eftpteyn, AmuUet de Vkeolt Normate (8) z. p. 106 (1898).
300 TRANSCENDENTAL FUNCTIONS. [CHAP. XH.
at the point < =» 0. The two formulae for On (^) can be united by reversing
the order of the terms ; thus
OnW«-^jrH-|l + 2(2n-2)"^2.4(2n-2)(2n-4)"*"-J'
the series terminating with the term in z^ or jf^K
We thus have Neumann's expansion
f(z) = Oo/o W + «i/i (z) + a, J, (z) + ...,
where ao=/(0)/
and On (n > 0) is twice the residue of 0» {t)f{t) at the point ^ = 0, so that
On^^^l On(t)f{t)dt,
y
where 7 is any simple contour surrounding the origin.
164. Proof of Neumanria expansion.
The method by which this result has been found cannot be regarded as a
proof, since the possibility of the expansion was assumed. We can, however,
now furnish a proof by determining directly the sum of the series obtained.
From the definition of On {z\ we can at once obtain the identities
0„+. {z) + 2 ^^ - 0„_, {z) = 0, (n > 0).
d tV
".W'-s©.
Writing the first of these equations in the symbolic form
On+x-2DOn-On.i-0, where i)-^,
and solving the series of recurrence-equations obtained by giving n integer
values, in the same way as if D were an algebraic quantity, we obtain for On
the symbolic expression
On(^) = i[{-i) + (i>+l)»}-+{-i)-(i>'+im(j)-
This symbolic expression can be transformed into a definite integral in
the following way.
164] BESSEL FUNCTIONa 801
1 r*
We have - - / e~*^ du,
t Jo
where the upper limit must be understood to mean that direction at infinity
which makes the real part of tu positive and infinite ; and therefore
or, writing tu = a?,
On (t) = f *i tr^' e'^[[w + (^ + ^«)*}« + {x^(a^ + <»)*}«] dw,
Jo
where the upper limit now means the real positive infinity, so that the
integration may be regarded as taken along the real axis of x.
Writing this in the form
<'-<«-r?rj:[r-i^T^<-'>-{^T(^/]-'^'
we have 0, (<) J, («) + 2 2 On (<) J,, («)
(by § 146)
« ^ Limit I 2^ < * xH»^+pM . e"* dx
^t x«oo Jo
y Limit /
t JT.flo Jo
^ — -«
6* " cia?.
In order that this integral may have a meaning, the real part of —r—
V
must be negative, a condition which is fulfilled when
If this inequality is satisfied, we have therefore
Oo{t)Jo(z) + 2 X On{t)Jniz) = :r^ .
«
From this result Neumann's expansion can at once be derived ; for let
/(z) be any function which is regulai* in the interior of a circle C whose
centre is at the origin, and let ^ be a point on the circumference of the circle.
Then if z be any point in the interior of the circle, the condition | jp | < | ^ | is
satisfied, and therefore we have
^ ^Oo(t)Jo(z) + 2iOn(t)Jn(z).
t-Z
n«l
302 TRANSCENDENTAL FUNCTIONS. [CHAP. XII.
Thus /(^) = ^— . y^—
•^ ^ 2'fnJc t — z
= 2^|^/(e)d^ jOo (0 Jo (z) + 2X0n{t) Jn (^)}
where flo =/(0)
and On = — . f On (0/(0 ^^ (^ > <>).
This establishes the validity of NeumaDn's expansion for points z within
the circle C,
Example, Shew that
cos f-^o W- 2/, (i)+2/4 («)-...,
166. Schldmilch*8 expansion of an arbitrary fwnction in terms of Bessel
functions of order zero.
Schlomilch * has given an expansion of a quite different character to that
of Neumann. His result may be stated thus :
Any function f(z) which is finite and continuous for real values of z
between the limits z=^0 and z^ir, both indu^sive, muy be expressed in the form
f(z)^ao + Oi Jo (-8^) + Oi Jo (2-er) + a^Jo (3-8^) + . . .,
where Oo = / (0) + - f ' w f \l - ^)-*/' (ut) dt du,
^Jo do
an^- I ucosnu I (1 - ^*)~*/' (ut) dtdu (n > 0).
^Jo Jo
Schl5milch's proof is substantially as follows
Suppose that F and /are two functions connected by the relation
f(z) = lf\l-^)-iF(zs)ds.
Then we have
2 p
f(z) = -[\l-'^)-^sr(zs)ds.
TTJo
♦ ZeiUehnftflT Math. u. Phynh, n. (1S67).
165] BESSEL FUNCTIONa 803
In this equation, write zt for z, multiply both sides by -er (1 — ^«)-* dt, and
integrate with respect to t between the limits ^ = and t^\. Thus
^ ['(1 - t^yif(zt)dt - — (\l - «•)-* dt l\l - l^)'^8F'(Z8t)d8
Jo IT J Jo
2 r» /•(*•-»•)*
where x^zst^ y^z8{l — t^)^.
Performing the integrations, we have
-/ {s^-a^''fy^F'{x)dxdy,
TT J Jo
f \l - e«)*/' (^) (ft = F{z) - J?'(0).
Jo
2: ,
'o
Now by the definition of the function/, we have
/(0)^F(0).
Thus F{z) =/(0) + z f\l - <«)-*/' (zt) dt.
Jo
This equation expresses the function F explicitly in terms of the function
/, whereas in the original definition / was expressed explicitly in terms
o{ F.
In order to obtain Schlomilch's expansion, it is merely necessary to apply
Fourier's theorem to the function F(zs). We thus have
f(z)^- ( (l-«»)-*(fojl ('F(u)du + - 1 rcosnwcosn^m)^^!
TT^o {jrJo ^ i»«lJo j
If' 2 * T'
= - I F(u)du + — 2 / cos nuF{u) Jo (nz)diu
^Jo ^ »»i Jo
In this equation, replace F{u) by its value in terms oi f{v). Thus
we have
+ - i J, (tl?) r cos nw 1/(0) + u f (1 -«•)-*/' (we) (ftl dw,
^ »-i Jo I Jo )
which is Schlomilch's expansion.
Example, Shew that if < f < ir, the expression
|*-2 |^,(*) +5^o(3»)+i^.(6*)+...}
804 TRANSCENDENTAL FUNCnON& [OHAP. XIL
is equal to z ; but that, if ir ^f ^ 2ir, its value is
«+2irC08-i--2(*«-ir*),
where cos'^ - is taken between and -x .
z 3
Find the value of the expression when » lies between 2«r and Zw.
(Cambridge Mathematical Tripos.)
166. TainJ^iXtion of the Bessel functions.
Many numerical tables of the Bessel functions have been published.
Meissel's tables (Berlin, 1889) give the functions Jo(z) and Ji{z) to 12
decimal places for real values of z from jer = to er = 15^, at intervals of 001.
Tables of the second solution F© (z), defined by the equation
z^ I 1\ ^
Y,{z)^J^{z)\o%Z'\r-^^''\\ + 27 2rT« ■*■•••'
from ^ s= to ir = 10*2, are given by B. A. Smith, Messenger of Math, xxvL
(1897).
The British Association Reports for 1889, 1893, 1896, contain tables of
the functions /n(^)> which are solutions of the differential equation
dhb \du f. ^n^ ^
so that In(^)^i'^Jn(i^)'
A table of the first 40 roots of Jo (^) is given by Wilson and Peirce, BtM,
Amer. Math. Soc. in. (1897).
Miscellaneous Examples.
1. Shew (ag. by multiplying the expansions for e^ ~<^ and «~^ " if , and equating
the terms independent of t) that
{^oWl'+S {^1 (*)}»+2 {y, «}»+2 {J, (»)}»+... = !,
and hence that, for real values of s, J^ {z) can never exceed unity, and the other Bessel
coefficients of higher order can never exceed 2~l.
2. Shew that, for all values of fi and v,
MISC. EXS.] BES8EL FUNCTIONS. 305
3. Shew that
4. Shew that
«^nW w+1- n-h2- w+3-../
5. Shew that
/_mW«/m-iW+«^-m+iW*^mW==
2sin/i9r
«-«
6. If v^-/! be denoted by Q^ («), shew that
(Lommel.)
dz z
7. Shew that
^_%ii).l_i(!LhI)e.w+,{e.W}«.
8. If the function
— I 2^co8'^t«cos(mu~;;sint«)(;?ti
JJ{z)= 2_ :^,(i«)'»^-m.*.p,
(which when k is zero reduces to a Bessel Amotion) be denoted by J^* (z\ shew that
where N^m, *. p is the " Cauch/s number ** defined by the equation
Shew further that this function satisfies the equations
and ^m'^\z)=2mJ^' (z)-^2(k+l){jLi{z)-j!^+i{z)},
(Bourlet)
9. If quantities v and i/'are connected by the equations
Af^E-eainEy cosi;=_ jz. where |6|<1,
«hewthat v=jr-h2(l-fl8)i 2 i (i«)»/^*(m«)-sinmJ/;
-where J^*(^)=- I (2costi)*cos(mti-;?8ini«)(;?t^
"■y
10. Prove that
P,-(cos^)=^^/^|(a.*+y«)i ^1^,
•where a«rcos^, ^+y*»»r*sin* $, and c„*» is a numerical quantity.
(Cambridge Mathematical Tripos, Part II, 189a)
W. A. 20
306 TRANSCENDENTAL FUNCTIONS. [CHAP. XII.
11. Shew that, if n is a positive integer and (m+2n+ 1) is positive,
(Cambridge Mathematical Tripos, Part I, 1899.)
12. Prove that
2 r*
Jo (2)=- I sin (« cosh «) fl?«.
(Cambridge Mathematical Tripos, Part II, 1893.)
13. Prove that
and if Y^ (z) is HankeFs second solution of Bessel's equation, defined by the equation
i r, (.) = Limit '^-.(^)-'^«(^)<=<^^ ,
T »=faitefer sm nir
■^••^ ?'--»-i=-5i?<Mi,('*£r'(^')-
14. Shew how to express ^J^ (z) in the form
AJ^ z)-{-BJo{z\
where A, B are polynomials in z ; and prove that
J4(6*)+3^o(6*)-0,
3./fl(30*) + 6^,(30*)=0.
(Cambridge Mathematical Tripos, Part II, 1896.)
16. Prove that, if •/„ (a{)=0 and J^ 03ft=O,
^^^xJ^{QX)M^x)dx^O, and |^*^{*^H(aa.0}«d:r-if8{^^^^(^)j8.
Hence prove that the roots of J^{x)=0^ other than zero, are all real and unequal.
(Cambridge Mathematical Tripos, Part I, 1893.>
16. Shew that
/* ^V~2~/
a:-*»+"»J«(ar)(ir=2-*+'«a'»-"»-i— --^— ^— ,
if 2w + l>f/i^-l.
(Cambridge Mathematical Tripos, Part I, 1898.).
MISC. EXS.] BESSEL FUNCTIONS. 307
17. Shew that
!- i '^{^^w
n pao
2 ^"ICiJ
(Lommel.)
18. Shew that the solution of the differential equation
where ^ and ^ are arbitrary functions of «, is
y-(^y<^-^^(^)+^-^-'W}-
19. Shew that
^^y /^V^(«sin^)sin'*+i^cW«2-*y«+j(«).
20. In the equation
the quantity n is real ; shew that a solution is given by
( - 1)"» z*^ cos {u^-n log z)
(Hobson.)
cos (n log «) - S
«-i 2»*m!(l+n2)*(4+n2)* (wi^+n^)* *
where t^^ denotes
tan-i?+tan-i5+...+tan-i-.
12 m
(Cambridge Mathematical Tripos, Part II, 1694.)
21. Prove that the complete primitive of the differential equation
where m is a positive integer, is
u^AI^{z)+BK^{z\
where, for real values of z^
^" ^=1-73^^1) /." *-«-'"*«'>l>'"*'^-
Prove also that
/eo
(««+««)-'»-Jco8Md«.
20—2
308 TRANSCENDENTAL FUNCTIONS. [CHAP. Xn.
Shew that for very small values of z,
jro(;^)--log|--677...,
and that for very large values of z,
(Cambridge Mathematical Tripoe, Part II, 1898.)
22. If C be any ciu*ve in the complex domain, and m and n are integers, shew that
j^O^{z)0^(z)dz^O,
where k^O \f the curve does not include the origin ; and, if the curve does include the
origin,
it=0 if m + w,
its=2»rt if m=n.
CHAPTER XIIL
Applications to the Equations of Mathematical Physics.
167. Introduction : illustration of the general method.
The functions which have been introduced in the three preceding
chapters are of very great importance in the applications of mathematics to
physical investigations. Such applications are outside the province of this
book ; but most of them depend essentially on one underlying circumstance,
namely that by means of these functions it is possible to construct series
which satisfy certain partial differential equations, known as the partial
differential equations of mathematical physics; and in this chapter it is
proposed to explain and illustrate this fundamental property.
The general method may be explained by considering first the solution
of the partial differential equation
^■^dj^=^ <1>'
a solution which, while resting on the same principles as those to be
developed later, does not require the use of any but the elementary functions
of analysis.
Consider any solution V{x, y) of this equation (1). Near any point at
which a branch of the function V{Xy y) is a regular function ctf x and y, and
which we may without loss of generality take as origin of coordinates, this
branch of the function V{x, y) can by Taylor's Theorem be expanded as a
power-series of the form
^(^» y) = ao+ Oifl;-!- 6iy -h o,^ -h 6a^ 4- Cay* + Os^:* + ;
on substituting this value of V in equation (1), and equating to zero the
coeflScients of the various powers of x and y, we obtain the relations
Oa + c, = 0,
3a8 + Cs = 0,
Sdj + 6, = 0,
310 TRANSCENDENTAL FUNCTIONS. ' [CHAP. XIII.
Fixing our attention on those terms in V which are homogeneous of the
nth degree in x and y combined, it is clear that the equalities just written
will furnish (n — 1) relations between the (n + 1) coefficients of these terms
of degree n. When these equations are satisfied, there will therefore remain
only {(n + 1) — (ti — 1)} or 2 coefficients really arbitrary in the terms of the
nth degree in F.
Now the expressions
and F=(ir — iy)*
satisfy equation (1), and therefore if A^ and B^ ate any arbitrary constants,
the expression
An {x + lyY ^Bn{x- iyY
satisfies equation (1), and is homogeneous of the nth degree in x and y, and
contains two arbitrary constants. It therefore represents the most general
form of the terms of the nth degree in V\ and so the general solution of
equation (1), regular at the origin, can be expressed in the form
F(a:,y) = ilo + ^(a? + ty) + 5i(a?-iy) + il,(^ + iy)« + A(«-iy)"+ (2),
where the quantities Aq, A^, B^, A^, ... are arbitrary constants.
This expansion furnishes the general solution of equation (1) ; what is
however in general needed is the particular solution of equation (1) which
satisfies some further conditions. As an example of the conditions most
fi'equently occurring, we shall suppose that the value of the required solution
V(x, y) is known at every point of the circumference of a circle, whose centre
is at the origin and whose radius is any quantity a ; it being supposed that
this circle lies wholly within the region for which V is regular. This being
given, we shall shew that the constants Aq, Ai, B^, ... can be found, and
the solution can be completely determined.
For writing
x = r cos 0y y^r sin 0,
the value of V is known when r = a, as a function of 0, say f(0). Let the
function /(^) be expanded as a Fourier series in the form
f(0) = Oo + Oi cos ^-h 6i8in ^ + Oacos 2^ + 63 sin 20 + (3),
where the coefficients a©, Oi, 61, a,, ... are given by the formulae
1 r*' \
•2ir
1 [^
an = - I /(t) COS ntdt
IT J Q
1 f^'
6n = — / f(t) sin ntdt
y (*).
168] APPLICATIONS TO THE EQUATIONS OF MATHEMATICAL PHYSICS. 311
Consider now the expression
ao + ^(oi COS ^ + 61 sin 0) + Q (a, cos 2^ + b^ sin 20) + (5).
This expression (0) reduces to (3), i.e. to /(^), when r=^a\ and since we
have
r^cosn0=^ {(x + iy)* + (a? - iy)»},
r« sin w^ = 2^ {(a? + iy)« - (a? - iy)**},
it is clear that the expression (5) is of the form (2), i.e. that it is a solution
of the equation (1).
It follows that the solution V of equation (1), which is characterised
by the condition that it has the value V=f(0) when r = a, is given by the
expansion
where
r fr\^
F= Oo + -(aicos ^ -h 61 sin ^) 4- -) (o^ cos 20 + 6, sin 2^) + ...,
a \af
I 1 f^'
<
1 f*'
a% = - I f(t) cos ntdt,
1 f^
bn = - I f{t) sin ntdt
\ ^ Jo
The principal object of this chapter will be to obtain theorems analogous
to this for the other partial differential equations of mathematical physics ;
the method followed will be in most respects similar to that by which this
result has been obtained.
168. Laplace's equation; the general solution; certain particular solutions.
The partial differential equation
a»F d^V d'V ^
da^ dy^ dz^
is known as Laplace's equation, or the potential-equation, and is of importance
in the investigations of mathematical physics.
The general solution of this equation was given by the author in 1902.
It may be written
r2ir
F= I f(x cos t-^-y sin t + iz, t) dt,
Jo
312 TRANSCENDENTAL FUNCTIONS. [CHAP. Xni.
where / is any arbitrary function of the two arguments x cos t + y sin t + iz
and t The solution is eflFected in Monthly Notices of the Royal Astron. Soc,
VoL LXii. In this chapter however we are -concerned not so much with the
general solution as with the particular solutions which satisfy certain further
conditions. To the consideration of these we shall now proceed.
Let the equation be transformed by taking instead of the independent
variables a?, y, z, a new set of independent variables r, 0, ^, connected with
them by the relations
x^rsmO cos ^,
y = r sin ^ sin ^,
z = r cos 0.
It is found without difficulty* that Laplace's equation becomes
drV dr)^ siu^e a<^« "^sin 0d0\^ dO)"^'
Let us seek for particular solutions of this equation, of the form
V^Re<P,
where R, 0, <I>, are functions respectively of r alone, alone, and ^ alone.
Substituting, we obtain
1 d f.dR\ 1 d ( , ^de\ 1 d^
Rdr\ dr)^(bHm0dd T^ ^ dff) "*" cDsm*^ d<^« ""'
Now the quantity
Rdr\ drj
does not involve ^ or ^ ; and since by this equation it is equal to
1 d / . ^d0\ 1 d^
@8in0d0V^ dd) 4>sin^^d<^«*
it clearly cannot vary with r : it is therefore independent of ?•, ^, and <^, and
so must be a constant ; this constant we shall write in the form n{n'ir 1).
^,(^f)-»(.-i)^-o.
We thus have
dr
Write r ^ 6**, so dr = e^du. Then this equation becomes
d / ., dR^
^U''£)-^<^^'^''='
d^R dR , , 1 \ z> A
or -FT +-3 n(n+l)R = 0.
du^ du ^
* The ^ork is given in full in Edwards' Differential Calctdut,
168] APPLICATIONS TO THE EQUATIONS OF MATHEMATICAI. PHYSICS. 313
This is a linear diflferential equation of the second order with constant
coefficients ; its solution, found in the usual way, is
where A and B are arbitrary constants.
The most general form of the function R is therefore
Considering next the function <I>, it can in the same way be shewn that
the quantity
is independent of r, 0, and (f>, and so must be a constant. Writing
this constant in the form — m', we have for the determination of <I> the
equation
~— 4- m*4> =
of which the general solution is
<I> = a cos m<f> + h sin m<^,
where a and h are arbitrary constants.
It thus appears that the expressions
r"cosm<^0 and r*sin77i^©
are particular solutions of Laplace's equation, if n and m are any constants
and is a function (of 6 only) which satisfies the equation
^ ^ %%\n0dd\ ddj sin*^
Writing cos^ = 2r, this becomes
© = 0.
But when m is a positive integer, this is (§ 129) the equation which is
satisfied by the associated Legendre functions of order n and degree m ; so a
particular solution is the function
Pn'^iz), or P„«»(cos^).
Hence generally we see that the (2n+ 1) expressions
r'^Pn (cos ^), r'* cos <^ Pn^cos ^), r~ cos 20 Pn' (cos ^), ..., r'^C0S7K/>Pn'*(C0S^),
r" sin <^ Pn"^ (cos 6), r^ sin 20 Pn^ (cos 0), ..., r^ sin iKf> Pn^ (cos 0),
where n is a positive integer, are particular solutions of Laplace's equation.
314 TRANSCENDENTAL FUNCTIONS. [CHAP. XIII.
Moreover, since PfJ^ (cos 0) is of the form sin"* ^ x a polynomial of degree
(n — m) in cos 0, it is easily seen that each of these quantities, if expressed in
terms of x, y, z, becomes a polynomial, homogeneous of degree n, in a?, y, z.
It can in fact be easily shewn, by using the result of § 132, that
r" cos m<f> Pn^ (cos 0)
is a constant multiple of
f
J Q
•2ip
(a? cos t + y sin. t-h iz)^ cos mt dt,
and that
r^ sin m<l> Pn^ (cos 0)
is a constant multiple of
/,
Sir
(x cos t + y sin t-^ izY sin mt dt,
from which their polynomial character is evident; these forms have the
further advantage of exhibiting these particular solutions as cases of the
general solution given at the beginning of this article.
Example, If coordinates r, d, ^ are defined by the equations
y = (r* - 1)* sin S cos (f),
.2= (r*- 1)* sin $ sin <^,
shew that the function
7= P^'» (r) P„»» (oos $) cos mit>
is a solution of Laplace's equation
169. jf^c series-solution of Laplace* s equation.
The particular solutions of Laplace's equation, which have been found in
the preceding article, enable us to express the general solution, in the form of
an infinite series involving Legendre functions. This series-solution will of
course be really equivalent to an expansion of the general solution
/,
Jir
fix COS ^ H- y sin ^ + u, t) dt
already mentioned ; but the series-form is (as will appear from § 170) more
convenient in determining solutions which satisfy given boundary-conditions.
For let V(Xj y, z) be any solution of Laplace's equation
da^ "^ dy^ "*■ dz^ "
170] APPLICATIONS TO THE EQUATIONS OF MATHEMATICAL PHYSICS. 315
Then in the neighbourhood of any ordinary point, which we may take as
the origin of coordinates, V can be expanded in the form
Substituting this expansion in Laplace's equation, and equating to zero
the coeflScients of the various powers of x, y, z, we obtain an infinite number
of linear relations between the coeflScients a©, Oi, 61, Ci, Oj, ....
There are ^n{n'-\) relations of this kind between the ^(71+ l)(n-h2)
coeflScients of terms of degi'ee n in the expansion of V: and so only
-{2 (n + 1) (n + 2) — s w (n — 1)1 or (2n + 1) of the coeflScients of terms of
degree n in the expansion of V are really independent. But in the last
article we have found (2n + 1) independent polynomials of degree n in a?, y, z,
which satisfy Laplace's equation, namely the quantities
r^Pn (cos 6\ r~ cos <\>Pf^ (cos 0), , r~ cos n<f>Pn^ (cos 0),
r^Bin (f>Pn^(cm0)y , r^ sin TK^Pn** (cos ^).
It follows that the terms which are of degree n in a?, y, z in the expansion*
of V must be a linear combination of these (2n + 1) quantities ; that is,
V must be expansible in the form
F = ilo + r [A^P^ (cos 0) + ^1^ cos ^P,^ (cos 0) + B,^ sin <I>P^^ (cos 0)]
+ r« {A^P^ (cos 0) + Ai^ cos <^P,» (cos 0) + A^* cos 20 P,' (cos 0)
+ Ba^ sin i^Pa^ (cos ^) + J8,« sin 20 P,^ (cos ^)} + . . . ,
where the quantities A^, A^, A-^y B^y ... are arbitrary constants.
170. Determination of a solution of Laplace's equation which satisfies
given boundary conditions.
In order to determine the unknown constants Aq, Ai^ A^y B^y ..., which
appear in the expansion just found, it is necessary to know the remaining
conditions which the function V is required to satisfy. A condition of
frequent occurrence is that V is to have certain assigned values at the points
of the surface of a sphere, which we may take as being of radius a and having
its centre at the origin. This sphere will be supposed to lie entirely within
the region for which F is a regular function of its argtiments a?, y, z. When
r = ay F is therefore to be equal to a given function f{0y <f>) of and 0.
The constants Aq, Aiy -4l^ JS/, ..., are therefore to be determined fix)m the
equation
f{0y ^)^Ao + a {ill Pi (cos 0) + ili» cos (f> Pi^ (cos 0) + A' sin <l>Pi^ (cos 0)]
+ a* {A^Pi (cos 0) + -4,^ cos (fyP^^ (cos 0)+ ...] + ...
316 TRANSCENDENTAL FUNCTIONS. [CHAP. XIII.
In order to obtain the value of one of these constants, say -4n'*, from this
equation, we multiply both sides of the equation by Pn** (cos ^) cos m^, and
integrate over the surface of the sphere. On the left-hand side we thus have
r i '/(^» ^) -Pn*" (cos d) COS nKf> sin dd d(f>.
Jo Jo
As to the right-hand side, we know that
/,
2ip
is zero except when r = m, and that
/•2»
COS m<f> cos r<f> d<l>
I COS m<f> sin r<f> dif>
Jo
is always zero ; and also (by § 130) that
f ' Pr"^ (cos 6) Pn"" (cos 0) siu d0
Jo
is zero except when r = n. It follows that on the right-hand side, every
term vanishes except the term
a'^^n"* re {Pn"* (cos 0)Y cos» m</> sin d0 d<f>.
Jo Jo
Since I cos' m^ d<f> = tt,
and (by § 1 30) /J {P„» (cos 0)}^ sin 6 d0 = ^^^ <^| .
this term has the value
2n-|- 1 (n — m)!
We have therefore the formula
^""^ = i^»' • (^! /o' ly^^- '^^ ^-^ <^°« ^> ""^^ "^<^ «^ ^ '^^ '^*'
which determines the coeflScients -4n"* in the expansion of V,
The coeflScients £„"* can be similarly determined : and so finally the
solution V of Laplace's eqxuition, which has the value f{0, <f>) at the surfa^ of
the sphere, is given for points in the interior of the sphere by the expansion
^ = Jo ^^ ^ (a)7«' ly^^'- *^'^ {^" ^"^^ ^^ ^" ^*'*'' ^^
+ 2 2 J^^^^;P„'»(co8 5')-Pn"(cosd)cosm(^-^')|8in^d^d<^'.
171] APPLICATIONS TO THE EQUATIONS OF MATHEMATICAL PHYSICS. 317
This result may be regarded as a three-dimensional analogue of the two-
dimensional result of § 167.
Elxample 1. Shew, by applying the expansion-theorem just given, that
P^ {cos ^ COS ^ + sin ^ sin ^ COB (<^ - <t/)} — P„ (cos 6) P„ (cos ff)
+ 2 J^ |J^; P,- (cos e) P,« (cos ff) COS m (* - <t>').
Example 2. Prove that if the product of a homogeneous polynomial of d^ree n in
0?, y, z and the function P„{cos^cos^+sin^sin^cos(^-^')} be integrated over the
smface of the sphere, the result is 4ir/(2n+l) multiplied by the value of the polynomial
at the point (^, ^').
(This can be proved by taking ^ to be zero, which involves no real loss of generality,
and expanding the polynomial by the theorem of this article.)
171. Particular solutions of Laplace's equation which depend on Bessel
functions.
It is possible to construct solutions of Laplace's equations in series in
several ways, of which that which has been given, and which depends on
Legendre functions, may be taken as representative. A fall discussion of
the other methods would be beyond the scope of this book, but a general
idea of them may be inferred from the result which will next be established,
namely that the Bessel functions furnish a group of particular solutions of
Laplace's equation, just as the Legendre functions do.
When Laplace's equation
da^ dy^ dz*
is expressed in terms of the " cylindrical coordinates *' z, p, ^, where p and ^
are defined by the equations
\x^ p cos <^,
[y = p sin ^,
it takes the form
dz^ dp" '^ pdp "^ p^d<l>^
Let us seek for particular solutions of this equation, of the form
F = ZP<I),
where Z, P, <I>, are functions of z alone, p alone, and ^ alone, respectively.
On substituting this value of F, Laplace's equation becomes
ld«Z l/d|P ldP\ 1 d«<l>
Zdz*"^ ?\dp''^ pdp)'^ p^d<f>^''
I
318 TRANSCENDENTAL FUNCTIONS. [CHAP. Xin.
This equation shews that the quantity
Z dz^
must be a constant independent of z, p, and ^ ; let this constant be denoted
by i^. Then on solving the equation
we have the particular solutions
Z = e** and Z—e"^.
Similarly the quantity
1 d^
<t>dif>^
is a constant, which may be denoted by — m' ; on solving the equation
we obtain the particular solutions
<I> = cosm<^ and <I> = 8inm^.
The equation to determine P is now
>-Jfn'- $)--«■
dp
On putting kp = y, this becomes BesseUs equation of order m,
*''+i^+ii
-?")'-»■
df y dy
a particular solution of which is
P = /m(y).
It follows that the expressions
e"^ cos 7n<f>Jm{lcp) and ^±** sin m^t/in (A;p),
where k and m are arbitrary constants, are particular solutions of LapUice's
equation,
172. Solution of the equation
— + ~ + 7=0.
We now proceed to consider another partial differential equation.
173] APPLICATIONS TO THE EQUATIONS OF MATHEMATICAL PHYSICS. 319
We have seen in the last article that Laplace's equation
da^ dy'* dz*
is satisfied by the particular solutions
^Jn(r) cos 710 and e* /„ (r) sin w^,
where a? = rcos^, y = r8in^.
But if we write TT^eT,
where F is a function of x and y only, the Laplace's equation for W becomes
It follow^ that, for all values of n, the quantities
Jn (r) cos nO and /» (^) sin nO
are particular solutions of this latter equation..
From these particular solutions, as in the case of the solution of the
equation
already described, we can build up the general solution of the equation
d^V 3'F
— + 4- F=0
00
in the form V= ^ J^ (r) {an cos nO + bn sin nO),
n=»o
where ao, Oi, a,, ..., 6i, b^, ..., are arbitrary constants.
173. Solution of the equation
3»F d*V a»F ^ ^
1 4- — I- F =
In order to solve the equation
^ a«F 3«F
which is likewise of great importance in the investigations of mathematical
physics, we first express the equation in terms of new independent vaiiables
r, 0, (f>, defined by the equations
a; = r sin ^ cos (f>,
y=^r sin sin (f>,
,z =rcos^,
320 TRANSCENDENTAL FUNCTIONS. [CHAP. XIII.
and theD endeavour to find particular solutions of the form
F=iee<i>,
where -B, ©, <I>, are functions respectively of r alone, 6 alone, and <^ alone.
Proceeding as in § 168, the diflFerential equation becomes
1 d/ d^N 1 d ( . ad^\ . 1 d^
■*■ 22 rfr V dr j "^ sin ^ d^ V®*" df>; '•' 4> ain» ^ d^» ° "•
This equation can be solved by the process used in § 168 for finding
particular solutions of Laplace's equation ; the quantity
^ RdrK dr)
must be a constant, which we shall denote by n (n + 1). If in the resulting
«quation
we write y = Rr^, it becomes
(. . !)1
which is BesseUs equation of order (^ + s) •
The quantity R can therefore be taken to be
R^r^Jn^^{r),
The equations for % and <I> are now found to be the same as those which
occur (§ 168) in the solution of Laplace's equation; and proceeding as in
§ 169, we find that the general solution of the partial differential equation
^ 9>F a»j
regular near the origin, can be expressed in Reform
F= i r-iJn^{r)
n=o
AnPn (cos 0) -h An' COS <t>Pn' (cOS ^) + . . . + An"" COS n<l>Pn'' (cOS 0)]
-I- Pn' sin <f>Pn' (cos ^) + . . . + 5„« siu TK^Pn** (cOS 0) )
+ Bn' si]
where the quantities A and B are arbitrary constants.
173] APPUCATIONS TO THE EQUATIONS OF MATHEMATICAL PHYSICS. 321
When a particular solution V of the equation is to be determined by the
condition that it is to take prescribed values at all points on the surface of a
sphere, the constants A and B are determined exactly as in § 170.
Example, Shew, as a case of the general expansion of this article, that
»sO
Note, The partial differential equations of §§ 172, 173, possess general solutions
analogous to that of Laplace's equation. The solution of the equation of § 172 is
where /is an arbitrary function ; and the solution of the equation of § 173 is
where / is an arbitrary function. For the proof of these results, reference may be made
to papers by the author.
Miscellaneous Examples.
1. If a solution V of Laplace's equation be symmetrical with respect to the axis of i,
and have the value V^f(z) at points on that axis, shew that its value at any other point
of space is
F-i f '/{«+» (^+y')* cos*} ^•
tr Jo
2. Deduce from the result of Example 1 that the potential of a circular ring of
mass M, whose equation is
IS
- f'if[c«+{«+i(a:»+y*)*co8<^}«]"*(i0.
^ J
3. Let P (x, y, ;r) be a point in space, and let the plane through P and the axis of z
make an angle ^ with the plane ix. Let this plane cut the circle whose equations are
in the points a and y, and let the angle aPy be denoted by $ and log (Pa/Py) by <r.
If <rj Bf 4>'be regarded as coordinates defining the position of the point P, shew that
Laplace's equation
^ a«F a«F
takes the form
d_ ( 8inh«r ?n . 1 [ sinh a 8 Fl 1 ^^o
9<r toosh«r-coe^ 8<rJ 8tf lco8h<r-cofl^ ^j sinh^ a (cosh a - cos $) d<f)^ ~ '
and that the quantities
F— (cosh o--coa tf)* cos «^ cos WM^i^_, (cosh c)
are solutions of it.
W. A. 21
CHAPTER XIV.
The Elliptic Function ff (z),
174 Introduction.
li f{z) denote any one of the circular functions sin z^ cobz, tau ^ ... , it is
well known that
f(z + 2,r) =/(^),
and hence that
f{z + 2n7r) =/(^),
where n is any positive or negative integer.
This fact is generally expressed by the statement that the circtdar
functions admit the period 2ir, They are on this account said to be periodic
functions; and in contradistinction to other classes of periodic functions,
which will be introduced subsequently, they are called singly-periodic
functions.
It will in fact be established in this chapter that a class of functions
exists possessing the following properties : if f(z) be any function of the
class, then f{z) is a one-valued function of z, with no singularities other than
poles in the finite part of the ir-plane ; moreover, /(-?) satisfies, for all values
of z, the equations
/(* + 2a,.) =/(«),
where cdi and ©j are two quantities independent of z. Functions f{z) of
this class are said to admit the quantities 2(!i)i and 2o>2 as periods, and are
called doubly-periodic Junctions or elliptic functions. The two periods 2(»i
and 2a>s pl^y ^^^ same part in the theory of elliptic functions as is played
by the single period 27r in the theory of circular functions.
By repeated application of the formulae written above, we obtain as the
characteristic equation of all elliptic functions the equation
f{z + 2mft), + 2no)8) = / W»
where m and n are any integers.
174, 175] THE ELLIPTIC FUNCTION ^{z). 323
176. Definition of |> {z).
The elliptic functions may^ as we have just seen, be regarded as a
generalisation of the circular functions. It is natural therefore to introduce
them into analysis by some definition analogous to one of the definitions
used in the theory of circular functions.
One mode of developing the theory of the circular functions is to start
from the infinite series
1 ** 1
z^ m^±i{z-miry'
It can be shewn that this series converges absolutely and uniformly for
air values of z except the values
-8^ = 0, ±7r, ±27r, ±37r...;
And that it admits the period 27r. If now its sum be denoted by (sin ^)~*,
and this be regarded as the definition of the function sin z^ then from this
definition we can derive all the properties of the function sin z^ and thus
a complete theory of the circular functions can be developed.
Similarly, as the basis of the theory of elliptic functions, we form the
infinite series
r^ + 2 {(^ - 2ma)i - ^nto^-^ - (2mft), -h 2n<»a)-«},
^here a*i and o), are any two quantities, independent of z, whose ratio is not
purely real, and where the summation extends over all integer and zero
{except simultaneous zero) values of m and of n.
It has been shewn in § 11 that this series is absolutely convergent for all
values of z, except the values -^ = 0, ± Wi, ± Wj, + o), ± o),, ± 2©! ± ©a, ... .
By comparing the series with the convergent series 2(tn* + n')~* as in
•§11, it is seen that this convergence is also uniform (§52). The series
therefore represent'S a one-valued function of z, regular for all values of the
variable z except the values z = ^mo)^ + 2no>, ; and at these points, which are
the singularities of the function, it clearly has poles of the second order.
We shall denote this function by the symbol fp^z). Its introduction is due
to Weierstrass.
There are other ways of introducing both the circular and elliptic fiinctions into
Analysis ; for the circular functions, the following may be mentioned :
(1) The geometrical definition, according to which sin z is the ratio of one side to the
hypotenuse, in a right-angled triangle of which one angle is z. This is the definition usually
given in the introductory chapter of treatises on Trigonometry : but from our point of
view it is defective, as it applies only to real values of z.
(2) The definition by means of the infinite product
sin«-z(l-5)(l-2gi)(l-^)....
21—2
324 TRANSCBNDBNTAL FUNCTIONS. [CHAP. XIV.
(3) The definition by the inversion of a definite integral,
We shall see subsequently that alternative definitions of the elliptic functions exist,
analogous to each of these definitions (1), (2), (3), and that they may if desired be taken
as fundamental in the theory.
Example, Prove that
••<"-<'K£.)'.L~~^('-^')'
176. Periodicity t and other properties^ of |> {z).
The function ^(z) is an even function of z^ Le. it satisfies the equation
For if —z be substituted for z in the series which defines |> {z\ the
resulting series is the same as the original series, except that the order of
the terms is changed. But since the series is absolutely convergent, this
change in order does not affect the value of the sum of the series; and
therefore we have
if>(^)=if>(-'^)-
Further, the function |f> {z) admits the quantity 2o>i as a period.
For
= (2: + 2(k)i)-» - r-« 4- S {(^r + 2a)i - 2ma)i - 2no>^y^ - (^ - 2mo)i - 2n<»a)"'}
= S {(2r - 2 (m - 1 ) Oh - 2na)8)-« -{z- ^nuo^ - ^am^)-^] .
where the last summation is extended over all integer and zero values of m.
and n without exception. But this last sum is zero, since its terms destroy
each other in pairs. Thus we have
Similarly f{z+ 2(k)a) = ^ {z\
and generally V{^ + imcoi -h 2rw«>g) = fp (z),
where m and n are any integers.
Therefore the function fp(z) admits the two periods 2ft)i and 2w2.
Differentiating the above results, we see that jf>' (z) is an odd function of
z, and admits the sams periods as ^ (z).
177. The period'paraUelograms,
The study of elliptic functions is much facilitated by a method of
geometrical representation which will now be explained.
V^PP
176 — 178] THE ELUPnc function ^ (z). 325
Suppose that in the plane of the variable z we mark the points z^O,
z s= 2(i(>x, z a 2(!i)9, z = 2a>i + 2a>s, ... and generally all the points comprised in
the formula z = ^ramx + 2na>8, where m and n are any positive or negative
integers or zero.
By joining the point ^ = by a straight line to the point ^ = 2o>i, then
joining the point 2o)i to the point 2(k>i + 2o)„ then joining the point 2o)i + 2(0,
to the point 2a>s, and lastly joining the point 2a>9 to the point ^ = 0, we
obtain a parallelogram in the ir-plane, which we shall call the fundamental
period-parallelogram.
It is clear that the whole ^-plane may be covered with a network of
parallelograms, which are each similar and equal to this parallelogram, and
which can be obtained by joining the other marked points by straight lines.
These parallelograms will be called period-parallelograms.
Then if < be any quantity, the points
z^t, ^ = t+2a>i, 5«5t+2a>2, ..., £: =» ^ -h 2tna)i + 27Mi),,
manifestly occupy corresponding positions in these parallelograms; these
points are said to be coryruent to each other.
It follows from the fundamental property of ^{z) that the fanctum f(z)
has the same valvs at all points which are congruent unth each other ; and
hence that the values which the function fp{z) has in any periodrparallelogram
are a mere repetition of the values which the function has in any other period-
parallelogram.
178. Expression of the function fp (z) by mea/ns of an integral.
We shall now obtain a form for |> {z) in terms of an integral, which will
be found to be of great importance in the theory of the function.
The quantity (p {z) - r^,
or 2 [{z — 2nK0i — 2na),)"* — (2mah -h 27k»,)~*},
is a regular fuuction of z in the neighbourhood of the point z^O, and is an
even function of z. It can therefore by Taylor's theorem be expanded, for
points z near the origin, in the form
where clearly we shall have
^ = 32 (2mo>i -h 2no)a)-^,
^ = 52 (2mo>i + 2wo>a)^.
Thus j>(^) = r-+g^ + g^ + ....
826 TRANSCENDENTAL FUNCTIONS. [CHAP. XIV.
Forming the square and the derivates of this expansion, we have
|»"(«)-6*-^ + ^ + |^,*»+....
Therefore jf** («) — s if>"(^) " n^'* **" **™"8 involving z* at least.
It follows that the function
is regular in the neighbourhood of the point z^0\ and as it is doubly-periodic
(for clearly any power or derivate of an elliptic function is likewise an elliptic
function) it must be regular in the neighbourhood of each of the points
z «s 2mcDi + 2n(k>s.
But the only singularities of ^{z) are at these points : and therefore the only
possible singularities of the function
are at these points. The latter function is consequently regular for all values
of z\ and so by Liouville's theorem (§ 47) is independent of z, and therefore
is equal to the value which it has at the point z^O, which is j^^s*
We have therefore the relation
P'(^)=^ii»"(^) + r2^..
Multiplying by 3 jf>' {z) and integrating, we have
where c is a constant ; on substituting the expansions in this equality, we
find that c = ^g^.
Thus, finally, the function p (z) satisfies the differential equation
jp'» (^) = 4 jf>> (^) - 5r, jp (^) - 5r„
where g^ and g^ (called the invariants) are given in terms of the periods
of jp(<j) by the equations
^r, = 60 2 (2ma)i + 2no),)~*,
5r, = 1402 (2?na)i + 2na),)"*.
178] THE ELLIPTIC FUNCTION fp(z). 327
This differential equation can be written in the form
where ^ »= j? (^)i
and therefore (since ^ (z) is infinite when z is zero) we have
which is the required expression of ^(z) in terras of an integral.
The preceding theorems may be illustrated by the results which correspond to them in
the theory of the circular functions. Thus we may in the following way discuss the
properties of a function f{z) (really ca$ec^z)f which we shall take to be defined by the
series
This series is clearly infinite at the points ^^O, ir, - ir, 2ir, ... ; for other values of z it
is absolutely and uniformly convergent, as is seen by comparing it with the series
H-l-«+l-«+2-a+2-«+3-«+3-*+....
The effect of adding any multiple of ir to ^ is to produce a new series whose terms are
the terms of the original series, arranged in a difierent order ; this does not affect the sum
of the series, since the convergence is absolute ; and therefore/ (^) is a periodic function of
X, with the period w.
By drawing parallel lines in the 2-plane at distances w from each other, we therefore
divide the plane into strips, such that at points occupying corresponding positions in the
different strips, /(f) has the same value. In each strip, f{z) has only one singularity,
namely at that one of the points 0, ir, — «-, 29r, -29r, ... which lies within the strip. The
function is not infinite at the infinite ends of the strip, because the several terms of the
series for f{z) are then small compared with the corresponding terms of the comparison-
senefi
1 +l-«+l-«+2-«+2-a+3-«+3-«+... .
Now near the point f =0, the function /(«) can be written in the form
/«='-*+'-'(l-i)"*+'-'(l+^)'*+(2')-'(l-2^) "*+...
=*-*+ir-*(l+l + 2-*+8-«+...)+»r-M(3+3+3. 2-« + 3. 2-«+. ..)+.. .
=«-»+2«-« . ^+,r-**». 3 . 2 . ^+...
Differentiating and squaring this equation, we have
It follows that
/"W-6/«(*)+4/(*)
328 1*RANSCENDENTAL PUNCTIONa [CHAP. XIV.
is a series containing no negative powers of z; it has therefore no singularity at the point
^^■■0, and therefore (since that is the only possible singularity) no singularity in the strip
which contains 2=0, and therefore (on account of the periodic property) no singularity in
any strip. It is therefore, by Liouville's theorem (§ 47), a constant : this constant must
be equal to the value of the function at the point ^--O, which (on substituting the expan-
sions) is found to be zero. We have therefore
/"W-6/«W+4/(,)-0.
Multiplying by 2/' (t) and integrating, we have
where c is a constant. On substituting the expansions, c is found to be zero, and therefore
/'»«=4/«(i){/(.)-l}
or (£)*-^ (^- ^)' ^^^ ^-/W>
which gives 2«- / f^{t- 1)"* dt
as the expreesioD of f(s) by means of an integral
Example, If y = ^ («), shew that
\dz) \ck)
where «i, «s, «} are the roots of the equation
For we have f(^('^)'^«^(')-9fi?(s)^9zf
and so (^y-4(y-«i)(y-«,)(y-6j.
Differentiating logarithmically, we have
''^-(y-«i)-^+(y-«j)-^+(y-«s)-^.
Dififorentiating again, we have
ds^ \dz*)
\dz) \aSJ
Adding the last equation, multiplied by ^, to the 49quare of the preceding equation,
multiplied by fj, we have the required result.
It may be noted that the left-hand side of the equation is half the Schwartzian derivative
of z with respect to y ; and hence the result shews that z is the quotient of two solutions
of the equation
179, 180] THE ELLIPTIC FUNCTION ff (z). 329
179. ITie homogeneity of the function fp (z).
When the Weierstrassian elliptic function is considered as depending on
its arguments and periods, it has a certain property of homogeneity, which
will now be investigated.
Let fp Iz, M denote the function formed with the argument z and periods
2a>i and 2^2. Then we have
It follows that the effect of multiplying the argument and the periods by
the earns quantity \ is equivalent to multiplying the function by X"^.
This relation can also be expressed in terms of the quantities gtt g^
For let fp(z; g^, g^) denote the function formed with the invariants
g^ and g^. Then we have
g^ss 60 2 (2mo)i + 2w<tt,)"*,
^fj =s 1402 (2mft>i + 2nG)a)~*.
The effect of replacing a>i and o>s by Xq>i and Xto^ respectively is therefore
to replace g^ and g^ by X^g^ and X~*5r, respectively; and thus we have
K';s„}.)-f{'. ^)
M"- ^
= X»jf>(X(r; X-*g„ \-*gt),
m
which expresses the homogeneity-property in terms of the invariants.
Example. Deduce the last result directly from the equation
180. The addition-theorem for the function fp (z).
The function fp(z) possesses an addition-tiieorem, i.e. a formula which
gives the value of |> (xr + y) in terms of the values of fp (z) and fp (y), where
JB and y are any quantities.
4f
•4
830
TRANSCENDENTAL FUNCTIONS.
[chap. XIV.
To obtain this formula, consider the expression
,1 F(*) 9'^')
! 1 «»(y) F'(y)
as a function of z.
Since it is compounded of doubly-periodic functions, it is itself a donbly-
periodic function ; and the only points at which it can have singularities are
the points at which the functions jp(^ + y) and ^{z) have singularities,
i.e. the points ^ = 0, z^^y, and points congruent (§ 177) with these.
Now for points z near the point ^ = 0, we can write the determinant in
the form
1 ip(y) + ^|)'(y) + W'(y) + ... -i?'(y)-^j?"(y)-...
^+Mfl^«^+ —
-2r-»+j^flra« + ..,
1 viy) ip'(y)
Expanding this determinant, we find that the terms involving negative
powers of z destroy each other; the determinant can therefore, in the
neighbourhood of the point ^ = 0, be expanded as a series of positive powers
of z ; that is, the function represented by the determinant has no singularity
at the point z^O; and therefore (by the periodic property) it has no
singularity at any of the points congruent with z^O.
Considering next the neighbourhood of the point z^ — y, write ^ = — y + a?.
The determinant can be written in the form
1 jf>(y) «>'(y)
and on expansion this is found to contain no negative powers of x. The
function represented by the determinant has therefore no singularity at the
point xr = — y or any of the congruent points.
The function has therefore no singularities, and so by Liouville's theorem
(§ 47) is independent of z. But it vanishes when z has the value y, since two
rows of the determinant are then identical.. The determinant is therefore
always zero.
We thus have the formula
1 ^{z + y) -p'(xr + y)
1 «>(y) v'iy)
= 0,
180]
THE ELLIPTIC FUNCTION ^{z).
331
true for all valnes of z and y. Since, by § 178, jf^ (^ + y)> §>' {^\ j?'(y) ^re at
once expressible in terms of jp(-f + y), jp(^), |>(y), respectively, this result
really expresses ^{z-k-y) in terms of ^{z) and jf>(y). It is therefore an
addition-theorem.
The addition-theorem may also be obtained in the following way.
Take rectangular axes Oxy Ou, in a plane ; and consider the intersections of the cubic
curve
with a straight line
The abscissae Xi, x,, x, of the points of intersection are the roots of the equation
<^(x)=0, where
<p (x) « {mx + n)« - 4c5 +^^ +^,.
The variation liXr in one of these abscissae, consequent on small changes dm and dn in
m and n, is therefore given by the equation
<f/ (Xr) 4x^+2 (mxr + n) (Xi^w + dn) ■■ 0,
whence
» to.
ft i x-dm + d»
— 2 2
Therefore
bO, by a well-known theorem in partial fractiona
8
Now when n is infinite, the abscissae x^, x^, x^ are all infinite : we may therefore
int^;rate the last equation over the series of positions of the straight line ytamx-^-ny and
obtain the result
3 /••
2 I {AXr^^-g^r-g^'^dXr^O.
r-1 y aL
If we write
*i-j?W, *a=j?(y)» *8-if>(t«^),
we have therefore z •\-y + tr ■■ 0.
But the ordinates of the three points of intersection are
«,-!>'(,), Wj-ip-Cy), «,-if)'(to).
Since the three points are collinear, we have
I ^3 W3 -0,
I jp(y) r(y)
which is the addition-theorem.
and therefore
332 TRANSCENDENTAL FUNCTIONS. [CHAP. XIV.
181. Another form of the addition-theorenL
The determinantal form of the addition-theorem given in the last article
may be replaced in the following way by a simpler, though less symmetrical,
formula.
Consider the equation
1 jp(a.) f(/(w) =0.
1 j?(y) jf>'(y)
If in this we replace fp> {x) by its value in terms of |> (x), and expand,
we have
{4t>» (w) ^g,fp (x) - g,\ { jp (z) - 1> (y)}«
= [fp' W {p (^) - j? (y)} + jf>' (y) {» (^) - 1> (^)}?.
This may be regarded as a cubic equation in the quantity |> (x). One of
its roots is jp(^) = |>(^+y), by the addition-theorem; and the other two
roots are jp (^) = jp (z) and fj>{x)^jp (y), since the determinant vanishes when
x^ or y is substituted for x. We have therefore
|> (z) + |p(y) + ip(^ + y) = Sum of roots of cubic
= - {Coefficient of p«(a;)} -r {Coefficient of f^(x)}
= i {J?' (^) - j?' (y)}M«> W - j? (y)}"*,
and thus we have
which is a new form of the addition-theorem.
Example 1. Prove that the expression
i{r W - r (y)}' {J? W - f (y)} -* - «» W - f («+y).
considered as a function of z^ has no singularities: and deduce the addition-theorem
for ip(z).
For the given expression, from the mode of its formation, can clearly have no singu-
larities except at the points e=xO, z=yy z^-y, and points congruent with these.
Consider then first the neighbourhood of the point z—0. The expression can be
expanded in the form
-P(y)-«r(y)--.
and this on reduction is found to contain no negative powers of e, the first non-zero term
being fp (y). The expression has therefore no singularity at the point ^=0.
181, 182] THE ELLIPTIC FUNCTION ^{z). 383
Considering next the neighbourhood of the point z^y^ we take fny+x ; the ezpressioQ
becomes
i {fr' (y)+*r (y)+ - - r (y)}Mf (y)+««>' (y)+ - - P (y)}-'- if (y) -*r (y) - -
-I»(2y)-xif>'(2y)-...,
and this on reduction is found to contain no negative powers of x ; there is therefore no
singularity at the point z^y.
The case of the point z^ — y can be similarly treated.
■
The given expression has therefore no singularities, and so by Liouville's theorem is
independent of z. But its value at the point z^O has been shewn to be ^ (y). We have
therefore, for all values of Zy
i{P''w-r(y)}*{i»W-jP(y)}-*-«>«-P(*+y)-«'(y)-o,
which is the addition-theorem.
Example 2. Shew that
P(«+y)+«>(«-y)- {if>«- jp(y)} -»({2PW (PW-igt) {|»W+i»(y)}-s'J.
For by the addition-theorem we have
Replacing fp^{z) by 4pW-5r,p(z)-^„ and replacing f^{y) by 4jf>8(y)-^,if>(y)-^3,
and reducing, we obtain the required result.
182. The roots e^, e,, e^.
Let n denote any one of the periods of g> {z), namely the quantities
2a),, 2a)3, 2q>i + 2ft)a> 2a)i — 2g)2> "- 2a)i — 2ca2, .... Then
§>' (^ fi^ = j>' Q ft - n) , since jf^ (^) has the period ft,
= — p' ^2 ft j , since |>' is an odd function of z.
It follows from this that unless 2^ ^^ itself a period (in which case
jp'T^ftj is infinite), jp'Uftj is zero.
We have therefore
|>'(«i) = 0, jf>'(«,) = 0, i?'(a>,) = 0,
where &>« stands for — (oh + ««>2)-
334 ' TRANSCENDENTAL FUNCTIONS. [CHAP. XIV.
Now denote the quantities jf>(a>i), |>(a>s), jf>(G)«) by ^, e,, «,, respectively.
Then the equation
jp'» (a)i) = 4|l* (a)i) - 5r, jp (o),) - ^r,,
or 0=4^»-5r8ei-5r,,
shews that ^i is a root of the cubic equation
Similarly e^ and ^s are roots of this equation.
Moreover, the quantities ei, e,, e, are distinct roots of the equation; for if
for example we had 61 = ^s, we should have fp (q>i) = fp (q>i), and therefore
o), = ± 0)1 + a period,
which is not the case.
We see therefore that the three roots of the cvhic
are ei, eg, c,, where
ei^ip (ft>i), «i = j? (wi), c, = p (o),),
and ft)i + ft>2 + ®j = 0.
The quantities ^1, es, ^ therefore satisfy the relations
ei + «, + es = 0,
_ 1
1
183. Addition of a half period to the argument of ^ (z).
From the addition-theorem we have
ip{z + G)0 + ip(2) + 61 =5|>'*(^) {jf>(^) - eij-«
= {if> (^) - (h] lip (z) - e,} {^ (z) - ex}-S
or j?(^ + fth) = ^ + («i - «i)(^ - <?i) (ipW - «i}"'.
This formula expresses the result of adding a half-period to the argument
of the Weierstrassian elliptic function.
183, 184] THE ELLIPTIC FUNCTION ^ (z), 335
Example 1. Shew that
is a multiple of the dlBcriminaut of the ec[uation
For we have
jf> (Z + »i) - «! = (61 - «jj) («! - «3) {P («) - <H}-1.
Differentiating, we have
Therefore
- 16 (ej - e^y (<52 - tfj)* (63-61)2,
which is a multiple of the discriminant of the equation
4 (^-61) (^—62) (^-6^=0.
Example 2. Shew that
{jP(2^)-«i}{j?(2«)-«J+{«>(2^)-«i}{i?(2*)-6,}+{P(2»)-eJ{p(2^)-63} = |>(«)-^
184. Integration of (oa?* + 46^;* + Qca^ + 4(fo? + eY^.
We shall now shew how certain problems in the Integral Calculus, whose
solution cannot be found in terms of the elementary functions, can be solved
by aid of the function |> {z).
Let the general quartic polynomial be written
f{x) = our* + Aba^ + 6ca? + ^dx + e.
Let its invariants* be
g^sioe — 46d + 3c*,
9t^
a b c
bed
c d e
= ace + 2bcd - c» - (wP - i'^e ;
let its Hessian be
A(a?)«(ac - 6«)a?* + 2 (ad - 6c)a;» + (ae + 2bd - S<^)a^
-4- 2(6e - cd)a? + (c6 - cP),
and let its sextic covariant be
t(^)^l {-/(^) A' (^) + A (^)/' (^)}
= (a"d - 3a6c + 26«)aJ«H- ....
* The student who is not already familiar with the elements of the theory of binary forms is
referred to Bomside and Panton's Theory of Equatiom^ where the invariants and oovariants
of the quartio are discussed.
336 TSANSCENDENTAL FUlfCTION& [CHAP. XI7.
Then it is known that
<• (^) = - 4A' {x) + g,f* {x) h (x) - (7,/« (x).
If we write « = — A (x)//{x), this relation becomes
i^{x)^f*{x)i^-g^-g,).
Now ^,M.)/-q-y (.)/(.) ^
and so (4«' — 5^^ — ^r,)"* cfe = 2 {/(a?)}"* cte.
Let Xq be any root of the equation /(w) =» ; then to the value a? = o!^
corresponds « » 00 ; and hence, if we write
z=r{f(x)}-idx,
we have 2z « / (4^ — g^ — fftY^ dt.
It follows that ths eqimtion
»{^z\ 9z. 9t)^-h{x)lf{x)
18 an integrated form of the equation
= / {cur* + 4iba^ + Qcaf^ + ^dx + e}-*(fo?.
z
Example I. Shew that (with the same notation)
P'(2«;ir,.fl',)-T<(*){/(x)}-i.
Example 2. Shew also that, if
then fp (z-i-y) and |>(^~y) are the roots of the equation
where F(Xy u)—(M^* + 26a««(47+w)+c(:F*+4PM+w*)+2</(4;+v)+e,
and H {x, u) is derived from h (x) in the same way as F{Xj u) from/(j?).
(Cambridge Mathematical Tripos, Part II, 1896.)
186. Another solution of the integration-problem.
The integration discussed in the last article may also be effected in the
following way.
As before, let
z=r{/(x)}-idx.
J xo
185] THE ELUPTIC FUNCTION |> (^). 337
where f{x) =^00* + Aba? + Qca? + 4dx + 6,
and let ^0 l>e a root of the equation /(ar) = 0.
Then, by Taylor's theorem, we have
f{x) = (^ - x,)f {x,) +\{x- x,yr M + g (^ - a:.yr M
+ i (^ - ^oYr" M
Writing (x — Xq)"^ = f» we have
/ w = i-* {/ ('t-.) ?• + \ f" i^o) ? + 1 r {?=.) r + i /"" (^ .
and 80 * = /*{/' (^.)(? + 5/"(^,)r' + 6/"'(^.)? + ^/""(^"*d?.
Writing f = 4 {/'(ar,)}-' ^, we have
" = /," {4^ + 5 /" (^o) ^ + ^ /' (^•)/"' (^•) ^ + 24 Te/'* C^') /"" (^ ' ^^-
Now take a new variable of integration «, defined by the equation
this substitution destroys the term involving the square of the variable of
integration in the denominator, and we thus have
where
5'. = ^/"'(«'.)-h/'(^'>)/"'(«'«)'
^. = h {/' (^o)/" (^•) /'" (^o) - \ /"• (^o) - 1 /'* (^•)/"" H •
It can easily be verified that these latter quantities are the same as the
invariants g^ and g^ of the last article.
We have therefore
and therefore ^ "= if^ l-^) "" 24 ^" ^^^*
?=4{/'(^.))-'jif>W-i^/"(^,
and finally x = x,^\f' (x.) jjp {z) - ^ /" (x,)} "' .
This last equation is the integral-equivalent of the equation
z=r{fix)}-idx.
w. A. 22
338 TRANSCENDENTAL FUNCTIONS. [CHAP. XIV.
It may be observed that
$>' (z) = (4s» - <7^ - g,)i = i/' (x,) [/ix)]i ?»,
and hence that
Example, Shew that the integrated form of the equation
'hdx,
where ^o ^ ^^7 constant (not necessarily a root of f{x))y and f{x) is any qnartic function
of X is
where (f> is the Weierstrassian elliptic function formed with the invariants g^ and g^
of/(^).
Shew further that
186. Unifomiisation of curves of genua unity.
The theorem of the last article may be stated somewhat diflTerently
thus :
If two variables y and x are connected by an equation of the form
y« = cwc* + 4iba^ + 6ca?* + 4(ir + e,
then it is possible to express them in terms of a third variable z by means
of the equations
y = i /' M <»' (^) {fr> (^) - r4 /" (^ " '
where f{x) = aa^ + ^ba^ + 6ca^ + 4da? + e,
Xq is any root of th^ equation f{x) = 0, and the function ^(z) is formed with
the invariants g^ and g^ of the quarticf{x) ; moreover ^ the quantity z is defined^
by the equation
z^\''[f{x)]-^dx.
Now y is a two- valued function of x, since the quantity
± (cwr* + 46ar» + 6ca^ + ifdx + e)*
186] THE ELLIPTIC FUNCTION jf>(^). 339
may take either sign ; and x is a four- valued function of y, since the equation
in X
aa^ H- 46^5* + ^ca? + ^dx + (e - y«) =
has four roots. But on referring to the equations which express x and y in
terms of z^ we see that x and y are one-valued functions of z. It is this fact
which gives importance to the variable z\ zis called the uniformising variable
of the equation
y* = oar* + 4iba^ + Qca^ + ^dx + e.
The student who is acquainted with the theory of algebraic plane curves will be aware
that curves are classified according to their genvs*^ a number which may be geometrically
interpreted as the difference between the number of double points possessed by the curve
and the maximum number of double points which can be possessed by a ciuve of the same
degree as the given curve. Curves whose genus is zero are called unicursal curves ; if
f(Xy y)=0 is the equation of a unicursal curve, it is known that x and y can be expressed
in the form
»
where <f> and ^ are rational functions of their argument ; since rational functions are always
one-valued, it foUows that the variable t thus introduced is the uniformiting variable for
the equation /(j;, .y)=0 ; i.e., although y is in general a many- valued function of x, and x
is a many-valued function of y, yet x and y are one-valued functions of z.
Considering now curves whose genus is not zero, let
f{x,y)^0
be a ciurve of genus unity. Then it can be shewn that x and y cau be expressed iu
the form
'x=<^) {z)
where and ^ are now elliptic functions of their argument z ; x and y are thus expressed
as one- valued functions of 2, and z is the uniformising variable of the equation /(^, y)=0.
This result is obtained by writing
where ^and O are rational functions of their arguments, and choosing ^and O in such a
way that the equation /(:>;, y)KO is transformed into an equation of the form
we can then wnte
and X and y will thus be expressed as one- valued functions of z.
When the genus of the algebraic curve
n^yy)=o
is greater than unity, the uniformisation can be effected by means of automorphic
functions. Two classes of automorphic functions are known by which this uniformisation
* In French genre^ in German OetchUcht.
22—2
c
340 TRANSCENDENTAL FUNCTIONS. [CHAP. XIV.
may be effected : namely, one which was first given by Weber in Qdttinger NackriclUen,
1886, and one which was first given by the author, Phil. Trans., 1898. In the case of
Weber's functions, the " fundamental polygon " (the analogue of the period-parallelogram)
is "multiply-connected," i.e. consists of a region containing islands which are to be
regarded as not belonging to it. In the case of the functions described in Phil. TVatis.,
the fundamental-polygon is "simply-connected," i.e. is the area enclosed by a polygon.
This latter class of functions may be regarded as the immediate generalisation of elliptic
functions.
Miscellaneous Examples.
1. Shew that
«>(^+y)-j?(«-y)=-rwr(y){i?W-i?(y)}-*.
2. Prove that
where, on the right-hand side, the subject of differentiation is symmetrical in «, y, and w.
(Cambridge Mathematical Tripos, Part I, 1897.)
3. Shew that
r'(^-y) r'(y-^) r'o^-«)
r(^-y) r(y-^) r(^-^)
P(^-y) j?(y-tz^) i?(t^-«)
=i^2 r'(^-y) r'(2/"^) f?"'(w-z)
iPi'-y) PCy-^^) ip{w-z)
11 1
(Trinity College Scholarship Examination, 1898.)
4. If
simplify the expression
where c^, e.^, e^ are the values of jf> (z) for which J>' («)==0.
(Cambridge Mathematical Tripos, Part I, 1897.)
5. Prove that
2{PW-«}{iP(y)-i?W}Mj?(y+^)-«}*{j?(y-^)-4*=o,
where the sign of summation refers to any three arguments z, y, w, and e is any one of the
quantities ^i, e2) ^s*
(Cambridge Mathematical Tripos, Part I, 1896.)
6. Shew that
£i?i^i)= _|PJK)-#> W.
(Cambridge Mathematical Tripos, Part I, 1894.)
7. Prove that
if> (2«) - p(a,,) = {j!)' (^)}-« {if> w- f> (K)P (F «- P (»»+K)}*-
(Cambridge Mathematical Tripos, Parii 1, 1894.)
MISC. EXS.] THE ELLIPTIC FUNCTION fp {z). 341
8. If m be any constant, prove that
^l r r e^{fp{z) - ft (y)} p (^) ^^^y
" 2^J; {i?«-^i}{i?(y)-^i} '
where the summation refers to the values of jp («) for which jp' (^) is zero ; and the integrals
are indefinita
(Cambridge Mathematical Tripos, Part I, 1897.)
9. Let
and let (s0 {x) be the function defined by the equation
where the lower limit of the integral is arbitrary. Shew that
_J^Js^ ^ »^(a+y)+<^^(«) , 4>'{a-y) + 4>'{a) _ <i>'{a'¥y)-<i>'{x)
*(A'+y)-*(«) </>(a+y)-*(«) </>(a-y)-*(«) *(«+y)-*(^)
<»-(a-y)-<^-(:r) ^
(Hermite.)
10. Shew that when the change of variables
is applied to the equations
rfw_. i^l — _=0,
2iy4-l+K
they transform into the similar equations
2i7'+l+^f
Shew that the result of performing this change of variables three times in succession is
a return to the original variables |, i; ; and hence prove that if | and i; be denoted as
functions of u by E(u) and F{u) respectively, then
where A is one-third of a period of the fiuictions E(u) and F{u),
Shew that • iJ^(t*)=^- j? (w ; gi> ffz\
12
where ^s-^p+lgl^, 5^3= -1-^^- 216^'
(De Brun.)
CHAPTER XV.
The Elliptic Functions snz, cnz, dnz,
187. Construction of a doubly-periodic function with two simple poles
in each period-parallelogram.
The function ^{z\ which has been considered in the previous chapter,
is a doubly-periodic function of z, with a single pole of the second order in
each period-parallelogram, namely at the point congruent with the origin*.
We shall next introduce a doubly-periodic function which differs from ^ (z)
in having two poles, ecwh simple, in every period-parallelogram.
Consider the series
/(z) = S [{^ + 2mfi)i -h (2n + 1) to^}-^ - {2m^, -h (2n -h 1) ao^}-^
- {2r -f- (2m -h 1) ft), -h (2n -h 1) ft)^}-* -I- {(2m -h 1) ft>i + (2n -f- 1) a>»}-^,
in which the summation extends over all positive and negative integer and
zero values of m and n.
When the modulus of (2ma)i -h 2nft)j) is large (and we may suppose the
series arranged in order of ascending values of |2ma)i-h2nft),|), the terms
of the series bear a ratio of approximate equality to those of the series
2 [- 5 [2mwi H- (2n -h 1) ft),}-« -h z {(2m -h 1) ft), -h (2w -f- 1) ft),!"*],
or — ^ 2 {2mft), -|- (2n -h 1) ft>a}
i-a..--- -.,-n.
Ol
2ma)i +(2n-|- I)ft)al
and these terms bear a ratio of approximate equality to those of the series
- 22rft)i X {27nft)i -h (2n -h 1) ft)^}-*,
which again bear a finite ratio to those of the series
S (2mft)i -h 2nft)a)~',
which was shewn in § 11 to be an absolutely convergent series'.
• In the network of paraUelograma described in § 177, the poles of ^ {z) are not within the
parallelograms, but on their bounding lines. We may however suppose the whole network
slightly translated so as to bring the poles within the parallelograms.
187, 188] THE ELLIPTIC FUNCTIONS sn Z, CD 2, dn z. 343
It follows that the sertes which represents f{z) is absolutely convergent
for all values of Zy except for the exceptional values included in the formula
z = ma>i + (2w + 1) (Wj, (m, w, integers)
for which the several terms of the series are infinite, and which have been
tacitly excluded from the foregoing discussion of convergence.
Moreover, since the terms of the comparison-series are independent of z,
the convergence is (§ 52) not only absolute but uniform.
By a discussion similar to that in § 176, we can shew that/(2r) is a dovhly-
periodic function of z, whose periods are 2(Oi and 20)3 ; it is an odd function
of Zy so that
f{z) = -n-z);
and its singularities are at the points
z = m<Oi + (2n + 1) wg,
where m and n may have any integer or zero values; these singularities
are simple poles, with the residues + 1. There are two of these singularities
in each period-parallelogram.
188. Expression of tJte function f(z) by means of an integral.
The singularities of f{z) in the fundamental period-parallelogram are,
as we have seen, at the points z^a)^ and ^ = (Wj + oj.
Consider now the neighbourhood of the point z = ©j.
Writing -8^ = ft)j -I- ic, we have
/(ft)j + a?) = — /(— 0)2 — a?), since / is an odd function,
= — /(2ft)3--6)j — a?), since 2ft)2 is a period,
= -/(6)2 - a?),
from which it follows that /(oij + a?) is an odd function of x ; the expansion
oi f{z) in ascending powers of x will therefore contain only odd powers of a?.
Now
f{z) » S [{a: + 2m(o^ -h (2n -h 2) to^]"^ - [2m(^ -|- (2n + 1) «,}-*
- {a? -h (2m + 1) Oh + (2w -h 2) ©3}-^ + {(2m -h 1) ©i + (2n -h 1) <o^'%
where the summation extends over all positive and negative integer and zero
values of m and w.
In this expression, replace all expressions of the form (-4 + x)"^ by
their expansions A"^ — A~^x -h A^^a^ — . . . , x being supposed small. A term
944 TRANSCENDENTAL FUNCTIONS. [CHAP. XV.
in x"^ will arise fix)m the pair of values (m = 0, n = — 1), and we thus
have
w
where 5 = 2 [- [2mwi + 2wg)j}-« + {(2m -h 1) ©i + 2/wtfa}-«] .
the summation being in this case extended over all positive and negative
integer and zero values of m and n. excluding simultaneous zeros in the
first term.
If now by means of this expansion we express the quantity
as a series of powers of x, it is found that the negative powers of x destroy
each other ; this quantity has therefore no singularity at the point z = Wa.
Consider next the neighbourhood of the point 2: = wi + ©j.
Writing 2: = ©i + oja + y, we have
/(©i -h 6), + y) = — /(— ft)i — o>j — y), since / is an odd function,
= —/(cDi-hwa— y), since (2fth + 2ft)2) is a period.
It follows that f((Oi + Wa + y) is an odd function of y ; its expansion in
powers of y will therefore contain only odd powers of y.
Now expanding/(ir) in powers of y, in the same way as /(^) was formerly
expanded in powers of x, we find that
/(z) l + B'y + Cy + ...,
where 5' = S [- {(2m - 1) ©i + 2r?6)2}-» -h {2mG)i + 2nft)j}-«],
the summation extending over all positive and negative integer and zero
values of m and n, excluding simultaneous zeros in the second term.
(Comparing this with the expansion of B, we have
so /(^) = -l-5y + ay + ...,
and, as before, the quantity
has no singularity at the point z — fOx-\- ci>a.
Now the points z = 6), and ^r = ©i 4- <»j are the only possible singularities
of this quantity in the period-parallelogram ; it has therefore no singularity
in the parallelogram, and therefore (since it is doubly-periodic) no singularities
189] THE ELLIPTIC FUNCTIONS 811 ^, cn 2r, dn ^. 345
in the whole -er-plane ; it is therefore by Liouville's theorem (§47) a constant
independent o( z, say A.
The function /(^) therefore satisfies a diflFerential equation
Replacing B and A by new constants k and /t, we can write this in the
form
/'»(^) = {^,-/'(^)}{;^,-/M^)};
SO that, Baf(z) is zero when z is zero,
-ris-")
"'a.-rt dt.
We see therefore that the odd doubly-periodic function f(z), which has
periods 2ft)i and 2(d^ and simple poles at all points congruent with ^ = oh
and -8^ = ft)i + 0)2, may he regarded as defined by the equation
cnz)(]^ wji w
where k and fi are constants depending only on cdi and Wj.
189. The function sn z.
The function f{z) discussed in the last two articles can be expressed in
terms of another function, which we shall denote by sn z, in the following
way.
Replacing the variable t of integration by a new variable «, defined by
the equation ks = fU, we have
zr^fil (1 - «»)-*(! - *»«>)-* d«.
Jo
Now define the new function sn -e by the relation
fifijjbz) = kanz;
then we have
runM
Jo
This last equation can be regarded as the definition of the function sn z
m terms of its argument z and the constant-parameter k, which is called
the modulus ; it is analogous to the definition of the function sin z by the
relation
rwnz
= / (1 -«»)-* ds.
346 TRANSCENDENTAL FUNCTIONS, [CHAP. XV.
From the equation
fif(jiz) = & sn ^,
it is clear that the function sn z has the same general properties as f{z)y
namely, it is an odd one-valued doubly-periodic function of z, with two poles
in each period-parallelogram, the distance between the poles being half of
one of the periods. The two periods will be connected by a relation, as they
depend only on the single constant h
190. The functions en z and dn z.
We now proceed to introduce two other functions, either of which may
be regarded as bearing to the function sn^ a relation similar to that which
the function cos z bears to sin z,
Csaz
Since z^\ (1 - «»)-* (1 - k"^)-^ ds,
Jo
dz
we have -r? c = (1 — sn* z)~^ (1 —k^ sn' z)~^,
a{snz) ^ ^ ^
or -T- (sn -^) = (1 - sn' z)^ (1 - A:" sn» z)K
Now sn ir is a one-valued function of z, so its derivate must be also a one-
valued fimction. It follows that
(l-sn'^)*(l-A'sn«2r)*
can have no branch-points (§ 46), considered as a function of z ; and therefore
either
(a) Elach of the quantities (1 — sn'^r)* and (1 — Ar'sn'^r)* is a function of z
which has no branch-points, or
(/8) The functions (1— sn'z)* and (1— A'sn'^)* have branch-points, .but
are such that their product has no branch-points.
Now the alternative (fi) could be true only if the functions (1 — sn' z)^ and
(1 — Ar'sn'^)* had their branch-points at the same places ; but this is not the
case, since (1— sn'^)* has branch -points at the places when sn'2: = l, and
(1 — A'sn'^)* has not. The alternative ()8) being thus ruled out, we see that
the alternative (a) must hold.
If now we write
en 2: = (1 — sn'^:)*,
dn;r = (l — i'sn'-e)*,
where it is supposed that each of these functions has the value unity when
sn z is zero, then since en z and dn z have no branch-points, and have definite
values at the point 2: = 0, it follows that the functions en z and dn z are one-
valued functions of z.
190, 191] THE ELLIPTIC FUNCTIONS 811 ^, cn z, dn z. 347
They obviously satisfy the relations
an^z + cn^z= 1,
l^sn^z + dn'2:= 1.
The functions sn z, en z, dn z are often called the Jacobian elliptic
Ainctions,
The function cos z is in the same way a one- valued function, although the occurrence of
the radical in (1 -sin'^)' might lead us at first sight to suppose that it possessed branch-
points.
191. Expression of en z and dn z by means of integrals.
We shall next find, for the functions en z and dn z, integral-expressions
similar to that found in § 189 for sn^^.
Diflferentiating the equation
cn*z = 1 — 8n*z,
we have en -^ -y- en -? = — sn .8^ en z dn -s^,
dz
so -7- en -? = — sn -g' dn z
dz
= - 1(1 - en- z) (*'« -h k" cn« z)]^,
where &'* = 1 — A;*.
Thus ifcn^ = ^ we have
d^ = - (1 - <»)-* (*'« + h^t")-^ dt,
and therefore (since en ^ = 1 when z = 0)
z^r (l-.t«)-*(A'»-hifc»t')-^(ft.
In the same way we can shew that
J- dn -? = — A^ sn z en z,
dz
and z=r (l-^)-*(t«-A/»)-*d«.
J dm
Example 1. If cs 2»cn ^/sn z^ shew that
J C8«
Example 2. If sd 2 -> sn zjdn ^ shew that
- {"^ \\ - h'H^yk (1 +i:»^)-i efe
i
348 TRANSCENDENTAL FUNCTIONa [CHAP. XV.
192. The addition-theorem for the function dn z.
We shall next shew how to find dna:, where
in terms of the sn, en, and dn, of y and z : the result will be the addition^
theorem for the function dn.
Suppose that y and z vary, x remaining constant, so that
^^ — — 1
* Introducing new variables u and v, defined by the equations
w = en ^ en y,
t; = sn ^ sn y,
we have
or
dv 1 . ^ d^
, -T- sn 2: en V dn V + sn V en 2r dn -er -y-
dv _d£^ ^ ^ ^ ^ dy
du du \ \ az
T- — en z sn y dn y — en y sn -8^ dn ^ -1-
dv __ sn 2r en y dn y — sn y en ^ dn ir
du Gnysazdnz—cnzsnydny'
From this we obtain the equations
(-T-) - 1 =A:*(sn'y — 8n*2r)'(cny sn ^dn 2r — cn^rsny dny)"*,
dv
v^u-T- = (sn y en y dn ^ — sn 2: en ^ dn y) (en y sn j dn 2: - en 2: sn y dn y)""^
(-r-j ■~(^"'^;7~) =(sn'y--sn'^)'(cny sn^dnz — cn2:8n ydny)"*,
and consequently
L /"^V -. 1 - /"^ V ( - ^^V
i^Kdu) k'^Kdu) V ^'dul'
or
*■(«-"*■)'—*•■©■■
This equation is the equivalent, in the new variables, of the equation
dz -
dy"
It is a differential equation of Clairaut's type, and its integral is therefore
ifc» (t; - -Mc)* = 1 - A: V,
where c is an arbitrary constant.
192] THE ELLIPTIC FUNCTIONS sn ^, CH Zy dn z, 349
Thus the equation
A» (sn 2r sn y — c en -^ en y)* = 1 — AV
must be equivalent to the equation
where c is some funetion of x.
To determine c in terms of x, put y = ; then we have
ifc«c* en* a; = 1 - Jk V,
which gives c* = dn"'^; = dn~* (z + y).
Now the integral equation can be written in the form
c» (1 — Jt» -f jfcj en' y en' ^) — ick^ sn y sn 2: en y en ^r + (A:* sn' y sn' 2r — 1 ) = 0.
Solving this equation in c, we have
_ A^'sny sn 2: en y en ^ t {Ar* sn'y sn'^r en'y en* ^—(1— Ar'-I-A:' cn'yen*2:)(A:'sn'y sn'^—1 )}*
" 1 — A* -h A* en* y en* ^
_A:*snysn2renycn^±dnydn^
^^ ^" 1-A* + A;*en*yen*^ '
Sinee
A:* sn* y sn* 2: en* y en' ^ — dn*y dn* = (1 — A;* + A* en* y en* 2:) (A:* sn* y sn* ^ - 1),
this equation ean be written
A:* sn* V sn* ^ — 1
or
4* sn y sn 2r en y en 2r T dn y dn ^ *
, . . ±dnydn^iA:*snysn2:enyen^
an \Z + y) = = — ^i — i z .
^ ^' 1 — A:»sn*ysn*z
The two ambiguities of sign in this equation remain to be decided.
Taking ^r = 0, it is seen that the first ambiguous sign must be + ; so
J . . dn ?/ dn ^ + A^ sn V sn 2: en V en ^
dn(2r + y)i= — ^ -:=— — f — ^ .
^ ^ 1 — &*sn*ysn*-2r
Now suppose that y is a small quantity ; expanding both sides in ascend-
ing powers of y, and retaining only the terms involving the first power of y,
we have
dnz-^-y-^dnz^dnz ±I^8nzcnz,
Sinee -j- dn ^ = — i* sn 2? en jg,
dz
360 TRANSCENDENTAL FUNCTIONS. [CHAP. XV.
it is clear that the ambiguous sign must be — . We thus finally obtain the
addition'theorem for the function dn, namely
, , . dn-edny— Ar^sn^sny cn^cny
Example 1. Shew that
^' ^ ^' 1 - ifc* an* y sn* «
Example 2. Prove that
, J „ 2dn2^
l+dn2z
193. The addition-ikeorems for the functions sn z and en z.
To obtain the addition-theorem for the function sn ^, we have
sn(ir + y)=±j^{l-dn«(«+y))»
Substituting for dn {z + y) from the result of the last article, this equation
after some algebraical reduction gives
sn
. . _ sn ^ en y dn y + sn y en 2: dn -gf
(^ + y) - ± - l^^Jc^^^^Jsn^ •
On putting y = in this formula, it is seen that the ambiguous sign is -h ;
we thus obtain the additton-theorem for the function sn, namely
I V _ sn -8^ en y dn y + sn y en 2r dn -?
sn(^ + y)- —-,^^——— .
Similarly for the function en z we obtain the addition-theorem
. . _ en ^ en y — sn 2r dn ir sn y dn y
^""^^■^^^ 1-A;«sn»^sn^y *
These results may be regarded as analogous to the addition-theorems for
the circular-functions, namely
sin (-3^ -h y) = sin z cos y + cos z sin y,
cos (-8^ + y) = cos z cos y — sin ^ sin y,
to which, indeed, they reduce when k is put equal to zero.
Example 1. Prove that
sn(z4-y)sn(«-y)=:; — rs — 5 — o ^ >
Example 2. Shew that
, 1 - en 2jj
1 H-dn 2«
193-195] THE ELLIPTIC FUNCTIONS SD Z, CU Z, du Z, 351
194. The constant K,
We shall denote the integral
r(i-eo-*(i-jfc»e«)-*ctt
by if; it is clearly a constant depending only on the modulus k. The
ambiguity of sign in the radical will be removed by the supposition that at
the lower limit of integration the integrand has the value 1.
From the equation
z
'o
we see that sn ir= 1,
and hence en if = (1 — sn^ K)^ = 0,
Example, Prove that
dnir=(l-ifc»8n«£')* = A?'.
196. The periodicity of the elliptic functions with respect to K,
It will now appear that the constant K is intimately connected with the
periodicity of the elliptic functions sn z, en z, dn z.
For by the addition-theorem, we have
sn -g: en X* dn X* 4- sn ^ en z dn 2r en z
sn{z + K):=
Similarly cn(z + K)=='-k' ~
l-k'sn'^zsn^K dnz'
, sn^:
and cln(^-|-ir) = ^
z
Hence sn {z + "IK) = j^ / ^ jl = - sn £,
and similarly en {z + IK) = — en -3^,
dn(2: + 2iO = dn2r;
and finally sn (z + 4ir) = — sn (^ + 2ir) = sn z,
en {z + ^K) = en z,
dn {z + 4i0 = dn z.
352 TRANSCENDENTAL FUNCTIONS. [CHAP. XV.
This 4iK is a period for the functions sn z and en z, arid 2K is a period for
the function dnz.
Example. If cs 2 = en z/an z, shew that
cszca(K-z)^t,
196. The constant K',
We shall denote the integral
The ambiguity of sign in the radical will be removed by supposing that
at the lower limit of the integration the integrand has the value 1.
Write « = (1 - ib"^)-*.
Then (i-^)-*=^(l-A:V)i, and (1-^^)"* = ^^-^^*^
and d« = (1 - k'H^)'^MHdt.
1
Therefore £" = - i\ (1 - «»)-* (1 - h^^Y^ ds,
1
and so ' K-\- iK' = [*(! ~ «>)-* (1 - A^^)-* ds,
Jo
or m(K + iIC)^Ty
whence dn(ir + iZ')=0 and en (ir + tir)= ± ^.
To determine the ambiguous sign in the last equation, we observe that
the sign of i must be understood in the light of the relation
(l-s')-*=^(l-A;'»<')».
which was used in the transformation ; putting
« = sn(Z + iir') = p «=1.
ihf
And so en {K + iK) = - -^ .
Example. Shew that cni(ir+iJr')=(l-t)(^) .
196-198] THE ELLIPTIC FUNCTIONS sn 2r, en z, dm, 363
197. The periodicity of the elliptic functions vrith respect to K + iK',
The quantity K introduced in the last article is of importance in
connexion with the second period of the functions snz, cn-gf, dnir.
For by the addition-theorem, we have
/ . IT . -ET/x sn<^cn (^+tX')dn(^ + tir') + sn(-K' + tZOcn<^dn^
Sn (j^ + iL + iJtL ) = ?— ;; j-i TTtt ^TrTl
_ dn-er
A; en 2:'
ik' 1
Similarly en (^ + ^ + iK') = - -r- ,
xkj sn £
and , dn (^ + iT + iK') = .
^ GHZ
By repeated application of these formulae we have
sn (z + 2^^ + 2tZ') = - sn-g:,
en (^ + 2ir + 2iK) = en z,
dn(-^ + 2ir + 2iir) = -dnr,
aod fsn(^ + 4ir+4tZ0=sn^,
cn(^-|- 4^+ UK') = Gazy
dn(^-|-4J5r+4tZ') = dn2r.
Hence it appears that the function en z admits the period 2K + 2iK'y and
the functions sn z and dn z admit the period 4}K + 4iir'.
198. The periodicity of the elliptic functions with respect to iK.
By the addition-theorem, we have
sn (^-1- iZ') = 8n (z+ K+ iK'-K)
^ sn (z ■¥ K + iK')cn K dnK - mK en {z '\- K + iK)dn(z + K '\'iK')
l-k'sn^K sn^ (z + K-^-iK)
1
^ ksnz*
Similarly we find the equations
^ k snz
dn (z +iK) = — i .
snz
w. A. 23
354 TRANSCENDENTAL FUNCTIONa [CHAP. XV.
By repeated application of these formulae we obtain
sn (z + 2iK') = an z,
en {z + 2iK') = — en -gf,
dn{z + 2iK')^--dnz,
and (Bn(z + ^K')^snz,
en (z + 4dK') = en z,
dn (z + 4dK') =^ dn z,
80 that the function sn z admits 2iK' as a period, and the functions en z and
dn z admit 4dK' as a period.
199. The behaviour of the functions sn z, cnz,dnz, at the point z = iK\
For points in the neighbourhood of the point 2f = 0, the function saz can
be expanded by Taylor's theorem in the form
snj? = 8n0 + 58n'0 + 2^sn"0 + ...,
where accents denote derivativea
Since sn = 0,
sn'0 = cnOdnO = l,
8n''0 = 0,
the expansion
becomes
sn'"0=.-(l + A;»), etc.
8nz ^ z "1(1 -{- 1(^) z^ + ... .
Hence
cnj? = (l — sn'z)*
= 1—2^ +...,
and
dn^ = (l-*»sn«^)*
and therefore
sn(^ + tX) = -^i--
=5^{l-l(l+A.)^+...f-^
1 . i + h'
199-201] THE ELLIPTIC FUNCTIONS 811^, cn z, dn z. 365
— i 2k^ — 1
and similarly en (z + iK") = nr- H gjr- i? + ...
i 2 — Jfc*
and dn(-2r + i-K'') = --+ — f— i« + ....
It follows that at the point z = iK\ the functions sn z, cnZfdnz have simple
poles, with the residues
1 _i
k' k' *'
respectively.
200. General description of the functions sn ^, en z, dn ^.
Summarizing the foregoing investigations, we can describe the functions
sn z, en Zy and dn z, in the following terms.
(1) snz is a one- valued doubly-periodic function of z, its periods being
4iK and 2iK\ Its singularities are ^t all points congruent with z^iK'
and z = 2-ff' -f iK* ; they are simple poles, with the residues Ar* and — k"^
respectively ; and the function is zero at all points congruent with ^ = and
z = 2K,
It may be obeerved that no other function than sn z exists which fulfils this description.
For if </>(;;) be such a function, then
^ {£) - sn «
has no singularities, and so by Liouville's theorem is a constant independent of z ; but it is
zero when 2=0, and therefore the constant is zero ; that is.
When ](^ is real and positive and less than unity, it is e€wily seen that K
and K' are real, and sn z is real for real values of z and purely imaginary for
purely imaginary values of z,
(2) en ^ is a one- valued doubly-periodic function of z^ its periods being
4iK and 2K-\- 2iK\ Its singularities are at all points congruent with z^iK'
and z=i2K -{-iK* \ they are simple poles, with the residues ik^^ and —ik~^
respectively ; and the function is zero at all points congruent with z==K and
z^ZK.
(3) dn ^ is a one-valued doubly-periodic function of z, its periods being
2K and 4dK\ Its singularities are at all points congruent with z = iK' and
z = ZiK' ; they are simple poles, with the residues — i and + 1 respectively ;
and the function is zero at all points congruent with z = K + iK' and
z^K^2iK\
201. A geometrical illustration of the functions snz, cnz, dnz.
The Jacobian elliptic functions may be geometrically represented in the
following way.
Let the position of a point, on the surface of a sphere of radius unity, be
defined by (1) its perpendicular distance p from a fixed diameter of the
23—2
356 TRANSCENDENTAL FUNCTIONS. [CHAP. XV.
sphere, which we shall call the polar axiSy and (2) the angle y^ which the
plane through the point and the polar axis (the meridian plane) makes with
a fixed plane through the polar axis.
Then if ds denotes the arc of any curve traced on the sphere, we clearly
have the relation
(day = p« {dylry + (1 - p»)-' (dp^.
Let a curve (SeiflFert's spherical spiral) be drawn on the sphere, its
defining-equation being
dy^ = kds,
where A? is a constant. We have therefore for this curve
and so if « be measured from the pole, or point where the polar axis meets
the sphere, we have
« = f ' (1 - /)•)-* (1 - A;»p»)-* dp,
Jo
or p = sn «,
the function sn being formed with the modulus k.
The rectangular coordinates of the point s of the curve, referred to the
polar axis and an axis perpendicular to it in the meridian-plane, are p and
(1 — p')*, and can therefore be written sns and cns\ while dn* is easily seen
to be the cosine of the angle at which the curve cuts the meridian. Hence
if K be the length of the curve from the pole to the equator, it is obvious
that sn s and en s have the period 4jK', and dn s has the period 2K.
202. Connexion of the function sn z with the fwnction p (z).
We shall now shew how the functions considered in this chapter are
related to the elliptic function of Weierstrass.
Let ei, ej, e^ denote the quantities ei, ^, ^, taken in any order.
Li the integral
z^\ I (a? - e^~^{x - ^)-*(a? - e^-^dx,
let the variable of integi'ation be changed by the substitution
ei-ej
a: = e^ +
Thus
z= {l-t')'^{(ei-ej)-b(ej--et)t'}'^dt,
Jo
or {ei - 6^)* -8^ = (1 - «•)H^ (1 - l(^t^)-^ dt,
Jo
202, 203] THE ELLIPTIC FUNCTIONS sn z, cn z, dn z, 357
where A* =
e*-e,-
This is clearly equivalent to the equation
We thus obtain the result that the function ^{z)yf(yrmed with cmy periods y
can he expressed in terms of the function snz by the equation
the function sn being formed with the modulus
\ei - CjJ
Example, Shew that this relation can be written in either of the forms
and p(,)=^*-^^5!i(fiZ^}.
l-dnM(^-«,)*4
203. Expansion of snz as a trigonometric series.
Since sn^r is an odd function of ^, admitting the period 4ir (which we
shall for our present purpose suppose to be real), it can by Fourier's theorem
be expanded in a series of the foim
, . irz , 27r£^ 7 . oirz
sn^r = 6, sm 2^ + 6, sm 2^ + 6, sin 2^ + ... ,
1 f*^
where (§82) ^'' " ^ I s^^ ^ sin
2K
This expansion will (§ 78) be valid for all points in the ^-plane contained
in a belt parallel to the real axis and bounded by the lines whose equation is
Imaginary part of 2r = ± iK\
since within this belt the function sn z has no singularities.
We have now to evaluate the integrals 6^. We shall follow a proof due
substantially to Schlomilch.
Let OARSCBQPO be a figure in the plane of a variable t, consisting of
the rectangle whose vertices are the points
0(t = 0), A(t^ 2K)y C(t^2K+ 2iK')y B{t = 2iK%
with a very small semi-circular indentation PQ around the point t = iK\ and
another small semi-circular indentation R8 round the point t = 2K -f iK\
358 TRANSCENDENTAL FUNCTIONS. [CHAP. XV.
Consider the integral
I sn < « *^ (ft,
taken round this contour.
Since the integrand is regular everywhere in the interior of the contour,
we have (§ 36)
\ +1 +1 +( +( +f +f +f =0.
J OA Jab J es j so J cb J bq J qp J po
Consider first the integral along the semi-circular indentation QP,
Writing t = i-fiT' + Re^, we have
Jqp Jw
irwt /.-« nrJT inr
qp
2
w irw
snz
2
IT
i ^T^ r"2
= T « *^ / (1 + positive powers of R) dO
ir% -
a
rvK'
= — r ^ ^^ J when R tends to zero.
k
Similarly we have
imt • rwK'
snte^dt^i-iy^e' ^ .
Jbs k
Since sn (z + 2%K') = sn z, we have
J CB J OA*
and since sn (z + 2iQ = — sn ir, we have
f =(-iyf , and f =(-l)'f .
J^tB J PO J 80 J BQ
We thus have
Now equate to zero the imaginary parts of this equation. Since
iiyt
sate^^ dt
203]
THE ELLIPTIC FUNCTIONS anz, CD 2, dn^.
359
is real when t is purely imaginary, there is no imaginary part arising from
JBQ JPO
Therefore
nrJST'
nrJT
(l^e^)j^mtBm^dt=le'^{l^(^iy}.
2K k
Writing
this equation gives
IT
{\-<f)KK='j^<f{\-{-\r]
or
and
6y = if r is even.
Thus finally we have the expansion of sn ^ as a trigonometric series,
sn^
_ 27r / g* irz g*
Example, Prove that
. ^irz (fi . ^irz
A
• • • I •
Stt ( q^ nz , q^ Zirz . o* bnz ]
^^=2?tri^"^2Z + ii73«>«2^ + l-^co8^ + ..,|.
Miscellaneous Examples.
1. Shew that
2. Shew that
3. Prove that
4. Prove that
6. Prove that
6. Prove that
ens
rdn*
dt
z
/"cn z
{l±cn(,+y)}{l±cn(.-y)}=J^^|Bg_
dn*«=
ir^-fdn 2g -hitr»cn2g
Hrdn2«
cn?
360 TRANSCENDENTAL FUNCTIONS. [CHAP. XV.
7. Shew that
^ , dnz-cnz
t^+dnz-k^cm"
8. Shew that
8n(*+iJr)=(l+i') S-(1-^)SD«, '
/
sn
9. Prove that
8111 [srn-i {sn («+y)}+8m-i {an («-y)}]=^j__-j_-^.
10. Shew that
r • 1 f / . M w / xn cn'y— sn*ydn*«
co8[sm-i {aD(r+y)}~8m-M8n(^~y)}]= i_^aD»IaD«y *
11. Shew that the quarter-periods K and iK' are solutions of the equation
where «=i!*.
12. Shew that the quarter-periods K and tX" are Legendre functions of the argument
(l-2ife»), of order -J.
13. Shew that
en jScny sn O— y)dn a+cn y en a sn (y- a) dn /9+cn a en j8 sn (a- /3) dn y
-l-sn — y) sn (y - o) sn (a— /3) dn o dujS dn y =0.
(Cambridge Mathematical Tripos, Part I, 1894.)
14. l{u+v+r+s=Oy shew that
i;^ sn t« sn V en r en « - /r> on t£ en t7 sn r sn « - dn t« dn v+dn r dn «=0,
iP^ sn t« sn i;- iP^ sn r sn «-|-dn i« dn V en r en « - en tt en vdn r dn tfasOy
sn t^sn V dnrdn «~dn ii dn vsnrsn «+cn r en «-ont«cni7sO.
(H. J. S. Smith.)
16. Shew that, if a>x>fi>yy the substitutions
x~y=(a— y)dn^t« and ar-y=(3-y)dn"*»,
where it* ■« (a — jS) (o - y) ~ *, reduce the integrals
j {{a-'X){x- p){X'' y)}-^ dx and | {(a-x)(a?-/9)(4?-y)}-*d:a?
respectively to the forms 2tt (o - y) " ^ and 2v (a - y) " * ; and deduce that, if w + v = JT,
1 - sn^ u - sn* v+ifc* sn* u sn* v=0.
From the substitution y= (a- .ir) (a?-/3) («-y)~^ applied to the integral above with the
limits p and a, obtain the result
W IT
1%! cos* ^ + 6i sin* d) - * c?d = /V* COS* ^ + ft* sin* d) - * <W,
where o^, 6] are the arithmetic and geometric means between a and b.
(Cambridge Mathematical Tripos, Part I, 1895.)
w
MISC. EXS.] THE ELLIPTIC FUNCTIONS Sn2r, CU Z, du Z. 361
16. Shew how to exprees
I
as an elliptic integral of the first kind, in the case when both quadratic expressions have
imaginary linear factors.
If z^r{(x+l)(x^+x+l)]-idx,
express x in terms of z by means of Jacobi's elliptic functions.
(Cambridge Mathematical Tripos, Part 1, 1899.)
17. The difierent values of z satisfying the equation cn3z=a are z^, ^, ... z^.
Shew that
9
3iH n en Zr+f* U cn«,.=0.
(Cambridge Mathematical Tripos, Part I, 1899.)
18. Shew that
en « 2ir ( q^ 1FZ q^ ZirZ . q^ birz 1
J = r^ {r" — C0S2rp.-T-^---iC0S^r^+^^— iCOB-r-p— ...}-.
auz JcK [l-q 2K 1-^ 2K 1-^ 2K j
19. Prove that
k'anz 2ir
dnz
20. Shew that
27r f V • «■« Q^ ' 37r2 . q^ . 6rrz ]
= 15- (i +y «« 2Z - d^ """ 2^ + 1 +? ""• iz - •••; •
J ir 2n ( q itz ^ (fi 2ttz . \
21. Prove that
sn'
/l+ifea/TrX 1« /»r\8\ V • ^«
fl-Hr>/tr\ 3^ /7r\»l V .
"*"t 2;fc3 \^2iS:; 2/fc3 \^2iry / 1-^ ^
2K
Zirz
sm — =>
2K
biFZ
sm
"^t 2it3 V2i:; 2)E3 V2ir; j i -^^" 2ir
(Cambridge Mathematical Tripos, Part II, 1896.)
22. Shew that
*« sn2 j= p (a - xK') + Constant,
where the Weierstrassian elliptic function is formed with the periods 2K and 2xK'.
23. Shew that the differential equation
g={Ji*8n«*-i(l +*»)}«
admits the general integral
i* = {8ni(C-«)cni(C-a)dni(C-«)}-*{i4 + i9sn«i(C-«)},
where A Mid B are arbitrary constants, and C^2K+%K\
CHAPTER XVI.
Elliptic Functions; General Theorems.
204. Relation between the residues of an elliptic function.
In this chapter we shall be chiefly concerned with properties of more
general elliptic functions than the special functions p (z), sn z, en z, and
dn^, which have been discussed in the two preceding chapters.
We shall first shew that the sum of the residues of any elliptic function,
with respect to those of its poles which are situated in any period-parallelogram,
is zero.
For let f(z) be an elliptic function, and let icji and 2(0^ be its periods.
The sum of the residues is, by § 56, equal to the integral
taken round the perimeter of the parallelogram.
Now in this integral, any two elements f(z)dz corresponding to congruent
line-elements dz on opposite sides of the parallelogram, are equal in magnitude
but opposite in sign, and therefore destroy each other. Hence the integral is
zero, which establishes the theorem.
The number of poles or zeros of an elliptic function contained within
a single period-parallelogram is often referred to as the number of irreducible
poles or zeros.
205. The order of an elliptic function.
We shall next shew that if c is any constant and f(z) is an elliptic
function, the number of roots of the equation
contained within a period-parallelogrami depends only on f(z), and is inde-
pendent of c, and is therefore equal to the number of irreducible zeros, amd also
to the number of irreducible poles.
204-206] ELLIPTIC FUNCTIONS i GENEBAL THEOREMS. 363
For the difference between the number of zeros of the function
m -
and the number of its poles, contained within the parallelogram, is (§ 60)
equal to the value of the integral
!«•]■
/'W
dz
2inj f{z)''C
taken round the perimeter of the parallelogram. But if P and Q are two
points congruent with each other, situated on opposite sides of the parallelo-
gram, then the elements /' {z) [f{z) — c]"^ dz arising from P and Q are equal
in magnitude but opposite in sign, and so destroy each other. The integral
is therefore zero; that is, the number of zeros of the function f(z)'-o
contained within the parallelogram is equal to the number of its poles, i.e. to
the number of the poles of 'f{z) ; but this latter number is independent of c,
which establishes the theorem.
The number of irreducible poles or zeros of an elliptic function is called
the order of the function. It must be noted that a zero or pole, which is
multiple of order n in the sense of " order " defined in §§ 39, 44, must be
counted as n zeros or poles for the purposes of this definition of " order."
The order is never less them two; for if an elliptic function had only
a single irreducible simple pole, the sum of its residues within any period-
parallelogram would not be zero, contrary to the theorem of the last article.
This explains why the functions discussed in the two preceding chapters,
which are of order two, are the simplest elliptic functions.
206. Expression of cmy elliptic function in terms of ^ {z) and jp' {z).
We shall now shew how any elliptic function can be expressed in terms
of the Weierstrassian elliptic function which has the same periods.
Let f{z) be any elliptic function, and let ^{z) be the Weierstrassian
elliptic function with the same periods 26t>i and 2q)s ; and let f^ {z) be the
derivate of ^{z).
First, we can write
f{z)-\ [fiz) + /(- z)] + Y^l^I^ ^' (^).
Now the functions f{z)+f{—z) and {/(^) — /(— -sr)} p'~*(^) are even
elliptic functions of z\ let {z) denote either of them : we shall now express
^ {z) in terms of (p {z).
Since <^(^) is an even function, it follows that if a be one of its zeros
in the fundamental period-parallelogram, then another of its zeros in the
parallelogram will be congruent to — a : its iiTeducible zeros may therefore
be arranged in two sets, say (h» (h>'"CLny and zeros congruent to
364 TRANSCENDENTAL FUNCTIONS. [CHAP. XVI.
Similarly its poles can be arranged in two sets, say 61, 62, ... 6n» and poles
congruent to — 61, — 6s, ... — 6n«
Now form the quantity
~4> (^) { J> (^) - P (6i)} {f> (^) - jf> (6,)} . . . {JP W - i? (6n)} '
where if one of the quantities Or or 6^ is zero, the corresponding factor
{pC-^) — j?(M} or { JP (^) -" j? (M) is to be omitted.
This quantity is a doubly-periodic function of ^r ; it clearly has no zeros
or singularities in the interior of the parallelogram, except possibly at ^^ = 0,
and therefore either it or its reciprocal has no singularities in the interior
of the parallelogram, and so has no singularities in the entire plane. It must
therefore by Liouville's theorem be a constant independent of z.
Thus
A (z^ - Constant x {<^^^>^ P fa)l W (^) - <f> («.)! » » » {P (^) - P K))
<^(.)- Constant x {p(,)_p(i^)j lj>(.)-|>(6,)} ... {f> (.) - jXtn)}"
The quantities {/(^) +/(--?)} and {/(^) -/(--?)} {f>'(^)l-* can thus be
expressed as rational functions of ^{z)\ and thus we obtain the theorem
that any elliptic function can he expressed in terms of the Weierstrassian
function formed with the same periods, the expression being linear in ff {z)
and rational in jf>(^).
Example. Shew that
snzcTkZ^uz^ ^k"^ ^{z- iK'\
where the function p' {z - iK') is formed with the periods 2K and 2iK'<,
207. Relation between any two elliptic functions which admit the same
periods.
We shall now shew that an algebraic relation exists between any two
elliptic functions whose periods are the same.
For let f(z) and <l>(z) be the functions. Then by the last article, f{z)
and (f> (z) can be expressed rationally in terms of jp (z) and jp' (z). Eliminating
jp(z) and p'(z) from the three equations constituted by
p''(z) =- i^ip^iz) ^ g,p(z) - g,
and these two relations, we have an algebraic relation between /(^) and <(> (z) ;
which establishes the theorem.
It is easy to find the degree of this equation in / and <}>. For if / be an
elliptic function of order m, and if ^ be of order n, then each value of/
determines m irreducible values of z, and each of these determines one value
207, 208] ELLIPTIC FUNCTIONS ; GENERAL THEOREMS. 365
of <^ : so to each value of / correspond m values of <^, and similarly to each
value of <t> correspond n values of /. The equation is therefore of degree m
in ^ and n in/
Thus IP {z) is of order 2, and jp' (z) of order 3. The relation between them, namely
rW=4pW-(73if>(^)-5r3,
should therefore be of degree 2 in fp'iz) and 3 in (p {z) — as in fact it is.
An obvious consequence of this proposition is that every elliptic function
is connected ,tuith its derivate by an algebraic equation.
Example. If ty u, v are three elliptic functions of the second order, with the same
periods and argument, shew that there exist in general between them two distinct relations
which are linear with respect to each of them, namely
Atuv-\-Buv+Cvt+Dtu+Et+Fu-^Ov+H=0y
A'tuv+B'uv+C'vt+iytu-\-E't+F'u+O'v+E'==0y
where A, B, ,,, , H* are constants.
208. Relation between the zeros and poles of an elliptic function.
We shall now shew that the sum of the affixes of the irreducible zeros of am,
elliptic function is equal to the sum of the affiles of its irreducible poleSy or
differs from this sum only by a period.
For it f(z) be the function, and 2a)i and 2a)8 its periods, the diflFerence in
question is (§ 59) equal to the integral
1 [ zf{z)dz
27rij f{z)
taken round the perimeter of the fundamental period-parallelogram. This
can be written
or
or
JL r f- {zfS') _ (20,. + ^)/' (26), + ^) ] ,
27riLJo t/{^)" /(2a„ + ^) J*"
+ p L '/'(^ + (2a>, H- y (2., -M) ^1
or
^{-^^'m^^'^'M
or 2«^~^**^'®S^"'"^*'''"Sl}.
366 TRANSCENDENTAL FUNCTIONS. [CHAP. XVI.
and as log 1 is zero or some multiple of 27rt, the last expression must be
either zero or some quantity of the form
A multiple of 2a>i + A multiple of 2a)a,
ie. a period This establishes the theorem.
Example, If F{£) is an elliptic function, for which ^, ^, ... are the irreducible poles,
and Ai, A^, ... the corresponding residues, and it f(z) is a one-valued function without
singularities in the parallelogram, shew that the integral
taken round the period-parallelogram, is equal to 2A^f(z^).
(Cambridge Mathematical Tripos, Part II, 1899.)
209. Thefimction f (z).
We shall next introduce a function f (5), defined by the equation
with the condition that ^{z) — z^^isto be zero when 2r = 0.
Since the infinite series which represents |> (z) is uniformly convergent, it
can be integrated term by term ; we thus have
f (^) = - /" [ir> + S {(-r - 2mfih - 2no>,)-* - {2nm, + Sno^)-*}] dz
= -2^* + S {(-r - 2ma>i - 2na>^)-^ + (2ma}i -{■ 2na),)-^ -f z (2mcih + 2nw,)"^},
since the condition, which ^(z) has to satisfy at2r=:0, is satisfied by this
choice of the constant of integration. The summation is, as usual, extended
over all positive and negative integer and zero values of m and w, except
simultaneous zero values.
When I 2ma>i 4- 27iq>, | is large (and we can suppose the series arranged in
ascending order of magnitude of | 2mo)i + 2na)s i)> the quantity
(z — 2mcDi — 2na)8)'"^ -f (2ma>i -f inaj^"^ -f z (2m<0i -f 2na)3)~*
bears a ratio of approximate equality to the quantity
- z^ (2?nft)i -i- 2yw»a)-».
The series which represents ^(z) can therefore be compared with the
series S (2nui>i -f 2n<»,)~*, and hence we see that it is absolutely convergent,
except at the singularities z = 2ma)i 4- 2na>2, ^^^ that the convergence is
uniform.
It is evident from the series that at its singularities z = 2mfi)i -f 2wg)2, the
function ^(z) has simple poles with residues unity; and that ^(z) is an odd
function of z.
209-211] ELLIPTIC functions; general theorems. 367
The function C (') niay be compared with the function cot ;;, whose expansion is
cot«=«->+ 2 {(«-wwr)"* + (i?Mr)->}.
The equation t- cot z = - cosec* z
corresponds to the equation
210. The quasi-periodicUy of the function f(«).
Since !>(•* + 2o),) = |> {z),
we have _ f (^ + 2a),) = ^ ? (4
or f(^ + 2w,) = ?(«) + 2i7„
and similarly K{^-^ ^Wa) = ?(-?) + 2i7„
where i/i and 172 are two constants introduced by integration.
Writing z^ — toi and 2r= — ©j in these relations respectively, we have
f (oh) = f (- «.) + 217, = - f (a>0 + 2i7„
ir («.) = f (- ««) + 2172 = - ?(a>2) + 2i;„
whence Vi^K (®i)>
which determines the constants rji and 17,.
If a:+y+«=0, shew that
{fW+f(y)+fW}*+r(^)+rcy)+rw=o.
(Schottky.)
This result may be regarded as the addition-theorem for the function ( {z),
211. Expression of an elliptic function, when the principal part of its
expansion at each of its singularities is given.
Lei f(z) be any elliptic function, with periods 26)i and 2a>2. Let its
irreducible singularities be at the points z=^a^, ctj, ... On; and let the
principal part of its expansion near the point ak be
z-au {z-auf '" {z-ajtYh'
Then if we consider the function
£?(^)=JJct,f(^-a*)-c^f'(^-a*) + ...+j^:^^^
d*
where f <*^ (z) denotes j- f (5), we see that
368 TRANSCENDENTAL FUNCTIONS. [CHAP. XVI.
(1) When z is replaced by {z + 2a>i), the function E{z) is increased by
n
n
But S Ck is zero, since the sum of the residues o( f(z) within a period-
Jb=l
parallelogram is zero. Hence E{z) admits the period 2(0i. Similarly E(z)
admits the period 2ft),. E (z) is therefore an elliptic function, with the same
periods ea/(z),
(2) Since the function ^^^^(z — ai) has singularities only at a* and
congruent points, and its principal part at a* is (— 1)*** ml (z — ajb)"**""*, we see
that E{z) has the same singularities as /(z), and the same principal parts
at them.
It follows from (1) and (2) that f{z)'-E{z) is a function with no
singularities in the whole plane; and therefore, by Liouville's theorem,
f(z) — E(z) is a constant. Thus the function f(z) can be expanded in the
farm
/(2r)=: Constant + 2 S J-^^' c*.f <-« (z - a,).
This theorem may be regarded as analogous to the decomposition of a rational function
into partial fractions, or the decomposition of a circular function into a series of co-
tangents (§ 76).
Example 1. Shew that
ir sn «= f («- liT') - f («- Sir- tiT') + Constant,
where the (-functions are formed with the periods 4 JT and 2iK\
1 pw rw
1 pw r«
Example 2. Shew that
j 1 »(y) lf^(y)
Extend this theorem to the case in which there are any number of variables.
(Cambridge Mathematical Tripos, Part II, 1894.)
212. The function a- (z).
We shall next introduce a function ^(z)^ defined by the equation
^log<r(^)=f(2r),
with the condition that a (z)lz is to be unity when z=^0.
212, 213] ELLIPTIC FUNCTIONS ; GENEBAL THEOREMS. 369
Since the convergence of the infinite series which represents ^(z) is
uniform, the series can be integrated term by term : we thus have
log a iz) = log . + 2 {log (l - ^^^ ^ g^ j
+
2?rwt)i + 2wa)a 2 (2ma)i + 27k»8)«j '
on choosing the constant of integration so as to satisfy the condition at ^ = ;
and therefore
the product being, as usual, extended over all integer and zero values of m
and n, except simultaneous zeros. The absolute convergence of this product
follows from that of the series
1 ^^ V 2ma>i + 2na>2 / "*" 2m^iT2n6^ "*" 2 (2m«i+ 2ni»j)*j '
which is established by comparison with the series
- 2 ^
3 (2ma>^ + 2n(o^y '
since the terms of the two series have ultimately a ratio of equality.
It is evident from the product-expression that a (z) is an odd function of
z, that its zeros are at the points z^2m(Di + 2na>s> and that z~^a'(z) tends to
the limit unity as z tends to zero.
The function <r (z) may be compared with the function sin 2, defined by the expansion
d
The relation ^ log (sin «) = cot «
corresponds to ^ log <''(«)" f (^)*
213. The qaasi'periodicity of the function <r {z).
On integrating the equation
f(^+2ah) = ?(^) + 2i;i
we have log a{z-\- 2<o^ = log tr {z)-^ 2%^ + Constant,
or ciz-^ 2o),) = C6*»»*^<r (z),
where c is a constant. To determine c, write ^ = — oh ; thus
<r (w,) = — c^"*'****^- (a)i),
or c = — e*»»*^,
W. A.
24
370 TRANSCENDENTAL FUNCTIONa [CHAP. XVL
and therefore <r (e + 2a)i) = - 6^i («+«.) a {£).
Similarly <r (^ + 2(0,) = - e*»«(*+«Oo- {z).
The behaviour of the function <r {z) when its argument is increased by
a period of (fi{z) is thus determined. By repeated application of these
formulae we can find the value of <r (^ + 2ma)i + 2n<»9), where m and n are
any integers.
An example shemng how the function o- {z) may he expressed as a singly-infinite product
We have
<r («) =zn (1 - ^ ?-« I ea»«,+ai»«,"*'^»m«,+2«-,)« ,
the Biunmation being extended over all positive and n^ative integer and zero values uf m
and n, exoept simultaneous zeros. This can be written in the form
»»±i V 27»«i/ ^.Ai \ 2w«8/
±00 • / » \ *
X
m~±l jissl \ 2w«i + 2w«)2/
n n (1+- ^ )ea»-i+2n4./»(2m«»+2iu.J«.
«i»±l n=l \ 2m«, + 27ki)j/
Now
±00
=.±l\ 2WKtt,/ IT 2«,
and n n fl-5 —; — )e2«i«i+an«i4"*"H2»i4#,+amJ
_«\ -.?«?izf
\ 2nu»J
^ 2mwi
^ sm^- = %• f -2««tg 1 1 *^ 1
= n ^<»i ^ ^—Ai 12««, (2m«,+am»,)"*"*(a«M.i+2ii^/
2<0j 29ta>2
Similarly
*« « / ^ \ ^* — +1 ^
n n ( 1+^5 I «a»t«i+2«Wi4 (2»n«i+2»M*a)*
m»±i «=i \ 2mtt>i+2w«2/
. (2nci>a + z)ir ^ « ^
^ sin^-— I ^— ^ f 2w»,8 +1 '^ \
sr n ^<^i ^ ^««±a8ti»«, (2iiu»,+2iM.^^'(2»iii»j+aiM^«;
-1 8in?^ 1+-^ * '
2a>j 2na>2
213]
ELLIPTIC FUNCTIONS; GENERAL THEOREMS.
371
Therefore
*• 1
2a»,
Sin
00
X n
«=i
3 . 1 ^^ sin ^ ^ ^' ^ *j / -2»M,r
Sin
2n<k)o7r
Sin ~^-^-
2a»i
(2mM,+2i»w9)
00
X n
»=i
e
z£
2a>,
+j ^ sin
1
27K04S'
sm '^
6
** r
2nMt' , . «*
(2mM,+2n«08) (2mw,+2n«tft)'
}
2a)
1
or
±m 1
«*JU ~ olu — ,_ L 5 z
2tt)i 2a>t g(2ii«,)« «=4i(2m«,+2»«,)«
Sin*'^
®1
Now write q^ e »! .
Then
.l„(^l^^.g!!fb±£); {,.,»-.-.g} {.. j!'-"'
}
. onfl»a7r
Sin" — *-
«1
{ 2ltoaii r|
m
1-25** 008 — hg^*
Now if the imaginary part of tojta^ is positive, we have | ^| < 1 ; and thus the infinite
product
1 - 2a*» cos — + ^*
«=i (i-g^**)^
converges ahsolutely, since the series
00
n=l
converges absolutely ; and hence we can separate off the exponential factors, and can
write
where (7 is a constant.
l-2^cos-- + a*»
«1
The quantity (7 can be very simply determined from the relation
«r(«+2«i)=-c^''»(*+"»)«rW;
for this gives
or
(7=-5>~
2<k>,
872 TRANSCENDENTAL FUNCTIONS. [CHAP. XVI.
We have therefore finally an expression for o- (2) as a singly-infinite product, namely
where q=€ •*! .
214. 2%6 integration of elliptic functions.
The integral of any elliptic function can be found in terms of the functions
f(«) and (r (z), by using the theorem given in §211, on the resolution of
elliptic functions into a sum of (['-functiona
In fact, in § 211 an expression
has been found for the elliptic function f(z) ; the indefinite integral of this
• •
expression is
c-^+ 2 c*. log o- (2: - aik) + 2 2 7 YTf <^*. f^*"*^'^^ - «*)>
*=i *=i #=2 V*— 17'
which is the required integral of f(z).
Example, The expression for fp^ {t\ found by the theorem of § 211, is
It follows that
jii^(z)dz=ifp^(z)+ ^g^ + Constant.
216. Expression of an elliptic furvction whose zeros and poles are knoum.
We have already seen (§ 205) that the number of irreducible zeros of an
elliptic function is equal to the number of its irreducible poles ; and that
(§ 208) the sum of the aflSxes of the zeros differs from the sum of the affixes
of the poles only by a quantity of the form {^nitoi + 2rko,), where m and n are
integers. By replacing the zeros and poles by others congruent to them, we
can reduce this difference to zero. Suppose this done, so that for a given
function f{z) the irreducible zeros are Oi, a,, ... On, and the irreducible
poles are 61, 6,, ... h^ where
Oi + Oj + . . . + On = 61 + 61 + . . . + 6n«
If any of the zeros or poles is multiple, of order k say, it will of course
be counted as if it were k distinct simple zeros or poles.
Consider now the quantity
^ / X ^ a{z-a^)a{z'-a^) a{z-a^
^ a{z — b^a{Z'-b^ a{z — bf^'
214,215] ELLIPTIC FUNCTIONS; GENERAL THEOREMS. 373
^ We have
Similarly J? (z + 2w^) = E (z).
Thus E(z) is an elliptic function, with the same periods aa /{z); and
therefore f{z)/E{z) admits these periods.
But the function f(z)/E(z) clearly has no zeros or poles at the points
djf eta, ... ttnj ^i> ••• Ofif
and so has no zeros or poles at any point of the 2r-plane. Therefore, by
Liouvilles theorem, f(z)/E{z) is a constant; and so finally
''' ^'^''a{z^b,)a{z-'b,) o-(ir-6n)'
where c is some constant.
An elliptic function is therefore determinate, save for a multiplicative
constant, when the places of its irreducible zeros and poles are known.
This is analogous to the factorisation of a rational function : if a rational function has
zeros at points a^ a,, ... o^, and poles at points &|, b^y ... b^, it can be expressed in the
form
(g-at) (g-ag) ... (z-On)
U^-b,)(z-b^,..(z-b^y
where c is a constant.
Example 1. Prove that
By differentiating this formula, shew that
and by further differentiation obtain the addition-theorem
J>(^+y)=-f>W-j>(y)+;{ ^g:g'g} V
Example 2. If
2 (aA-6A)=0,
shew that S «^ («A- ft.) ... o- («A- 6a) ■■■ .r (ax - 6^^^
A=l (r(OA— a,)... 1 ...<r(ax-o»)
24—3
374
TRANSCENDENTAL FUNCTIONS.
[chap. XVL
Miscellaneous Examples.
1. Shew that, ifp denote one of the functions an e, cnZf dn z, and if q and r denote the
other two, it is always possible to choose constants a, 6, c, such that
/
jxiz = a log (bq +cr).
2. Shew that every elliptic function of order n can be expressed as the quotient of two
expressions of the form
where 6, Oj, a,* ••• ^n* ^^ constants. (Painlev^.)
3. Prove that
ip(^-a)if>(^-6)-if>(a-6){if>(2-a) + fr>(^-6)-i?(a)-if>(6)}
+ j?(«)P(&).
(Cambridge Mathematical Tripos, Part II, 1895.)
4. Shew that
<r (x-k-y+z) a (x-y) <r (y - z) <r {z - a;) _1
1 i?(^) rw
1 j?(y) r(y)
Obtain the addition-theorem for the function fp (z) from this result
5. Establish the identity
1 PW i»'(^x)...««-"(^i) ^^^" ^"^
1 if>W jf>'W...««-*>(0 .
where the product is taken for all int^er values of X and ft from to n, with the restriction
6. Prove that
f(-?-a)-f(2-6)-f(a-6) + f(2a-26)
<r {z^2a-\-b) a {z- 2b-{-a)
<r{2b — 2a) <r(z-a) o-(z — b)'
(Cambridge Mathematical Tripos, Part II, 1896.)
7. Shew that, if Zf^+Zi+Z2+z^=0y then
{2f (z^)}'=3 {Sf (2a)} {2fp (2a)} + 2P' (2a),
the summations being taken for X=0, 1, 2, 3.
(Cambridge Mathematical Tripos, Part II, 1897.)
MISC. EX8.] ELLIPTIC FUNCTIONS ; GENERAL THEOREMS. 375
8. Prove that
f^^ (r(Z'^Zi)(r{z-¥22)<r(z+z^)fr(z+z^)
is a doubly-periodic function of z, such that
= ~2a-{i(«8+28- 2,-^4)} <r{i(«3+2i-«,-«4)}(r{i(«i+28-a3-«4)}.
(Cambridge Mathematical Tripos, Part II, 1893.)
9. If /(z) be a doubly-periodic function of the third order, with poles at «=0i, z^Cj,
-?=c„ and if ^ («) be a doubly-periodic function of the second order with the same periods
and poles at 2= a, 2= /3, its value in the neighbourhood of 2= a being
<(,(£)= A-+Xj(2-a)-|-X2(^-a)*-H...,
prove that
i X« {/" (a) -/" (/3)} - X {/' (a) +/' 03)} 2 * (c,) + {/(a) -/(/3)} {ZXk, + 2 <^ (c,) <^ (c,)} -0,
(Cambridge Mathematical Tripos, Part II, 1894.)
10. If X(;s) be an elliptic function with two poles a^ a,, and if ;?i, ^, ... z^^ be 2n
arbitrary arguments such that
shew that the determinant whose nth row is
1, X(«<), X«(^<),...X-(«<), \{Zi\ \(zi)\{Zi\ X«(^)Xi(^,),...X-2(z<)Xi(z<),
where \(^)=^X(«<),
vanishes identically.
(Cambridge Mathematical Tripos, Part II, 1893.)
11. Shew that, provided certain conditions of inequality are satisfied,
\ f\ e "I =5— (cot— --hcot-^)-h — Sg^'^sm — (»iw+»y),
where the summation applies to all positive integer values of m and n.
(Cambridge Mathematical Tripos, Part II, 1895.)
12. Assuming the formula
'W=* ' • V "" 2i, ? — (TTj^i — >
prove that
on condition that
(Cambridge Mathematical Tripos, Part II, 1896.)
376 TRANSCENDENTAL FUNCTIONS. [CHAP. XVI.
13. Shew that
26
1
^W-6(«-6)
(Dolbnia.)
INDEX OF TERMS EMPLOYED.
( The numbers refer to the pages, where the term occurs for the first time in the
hook or is defined.)
Absolute convergence, 12
,, valne (modulus), 5
Affix, 6
Analytic function, 46
Argand diagram, 6
Associated Legendre functions, 281
Asymptotic expansion, 163
Automorphic functions, 389
Beraoullian numbers and polynomials^ 97
Bessel coefficients, 266
„ functions, 274, 294
Branch, branch-point, 66
Circle of convergence, 29
Coefficients, Bessel, 266
Complex numbers, 4
Conditions, Dirichlet's, 146
Congruent, 325
Contiguous, 260
Continuation, 57
Continuity, 41
Contour, 47
Convergence, 10
absolute, 12
circle of, 29
„ radius of, 29
semi-, 12
uniform, 73
Cosine series, 138
Definite integral, 42
Dependence, 40
Derivate, 51, 53
Determinants, infinite, 35
Diagram, Argand, 6
Dirichlet*8 conditions, 146
„ integrals, 191
Double-oircnit integrals, 258
Doubly-periodic, 322
♦»
ft
I*
}>
i>
»»
♦»
»»
Elliptic function, 322
Equation, associated Legendre, 231
Bessel, 269
hypergeometric, 242
Laplace's, 311
Legendre, 206
Essential singularity, 63
Eulerian integrals, 184, 189
Expansion, asymptotic, 163
Exponents of a singularity, 245
Fourier series, 127
Function, analytic, 45
associated Legendre, 231
automorphic, 339
Bessel, 274, 294
Gamma-, 174
elliptic, 322
hypergeometric, 242
identity of, 59
Legendre, 209, 221
many-valued, 66
n
»♦
II
11
II
II
II
II
Gamma- function, 174
Genus, 339
Geometric series, 13
Hypergeometric series, 20, 240
„ function, 242
Identity of a function, 59
Infinite determinants, 35
products, 81
series, 10
II
i»
Infinity, point at, 64
Integrals, definite, 42
Dirichlet's, 191
double-oircnit, 258
Eulerian, 184, 189
Invariants, 326
II
II
II
378
Irreducible, 362
Kind of Legendre functions, 209, 221
„ Bessel „ 274, 296
Laplace's equation, 811
Legendre associated functions, 231
equation, 206
functions, 209, 221
polynomials, 204
Limit, 8
INDEX.
»i
i»
i»
Many-valued function, 66
Modulus, of complex quantity, 5
„ Jacobian elliptic functions, 346
Non-uniform convergence, 73
Numbers, BernouUian, 97
„ complex, 4
Order of Bessel coefficients, 267
„ functions, 274
elliptic functions, 368
Legendre functions, 209
„ polynomials, 204
pole, 63, 65
zero, 55, 64
>»
a
»»
i»
ti
II
Parallelogram, period-, 825
Part, principal, 63
Period, 322
Period-parallelogram, 325
Point, regular, 45
,, representative, 6
„ singular, 45
Pole, 68, 66
Polynomials, BernouUian, 97
„ Legendre, 204
Power-series, 28
Principal part, 68
Process of continuation, 57
Products, infinite, 81
Quantity, complex, 4
Radius of convergence, 29
Regular, 45, 46
Residue, 83
Representative point, 6
Semi-convergence, 12
Series, Fourier, 127
geometric, 18
hypergeometric, 20, 240
infinite, 10
power-, 28
sine and cosine, 188
Simple pole, 63
Sine series, 138
Singly-periodic, 322
Singularity, 45
„ essential, 68
of hypergeometric equation, 245
II
If
II
II
II
II
Uniform convergence, 73
Uniformisation, 388
Value, absolute (modulus), 5
Zero, 65, 64
CAMBBmOE ; PRINTED BY J. & C. V, CLAY, AT THE UMIVBBSITT PRB88.
fl t w
'•■^1 IF